I'm firmly convinced that the Martha we got on the telly does have a consistent character that evolves in a credible fashion as a result of her experiences with the Doctor, that a clear line can be drawn between the enthusiastic medical student we meet in Smith and Jones and the person who can play nuclear brinksmanship with Davros in Journey's End; the rest of this post constitutes an attempt to explain that. I'm not saying anyone has to agree with me, just detailing why I think what I think. (It's a tiny little bit picspammy: there are about eight smallish images beneath the cut, but nothing that should break people's connections too badly.)
Assumptions, Biases and Credits
Just a few bits and bobs to explain where I'm coming from.
Scope: I'm going to restrict myself to Martha's TV stories. The spinoffery's lovely (and there's so much of it!) but is constrained by having to fit between things. Some of the characterisation in the dinky little books in particular is lovely (oh, Pirate Loop and Last Dodo, I love you so), but pretty much by definition none of it is vital to the progression of her arc. By the same token, I'm not going to spend masses of time on Infinite Quest (while it was shown on telly, the whole thirteen tiny parts thing means that it's firmly set in the "generic Ten and Martha travelling together" time that we didn't actually see much of in S3). I'll be concentrating more on S3 than Torchwood and S4, because in the end it is S3 that contains Martha's main arc.
Author, Author: I'm not a big believer in authorial intent: it can be interesting, but I don't find it hugely relevant in the end. What's on the screen is what's on the screen, whatever the reasons behind it. I also don't believe we know as much about authorial intent as some people think: there are multiple creative inputs into any TV programme: actors, directors, set designers, etc. as well as the writer. And in the particular case of Who, even stated authorial intent is fairly useless IMO: RTD is a lying liar who lies, and I think Writer's Tale fairly comprehensively proves that his actual thought processes are much more complex than the soundbites he uses when he's in salesman mode in interviews and on Confidential.
Shipping: One of the traps that both Martha's fans and detractors fall into (I certainly do it often enough) is looking at everything about her through the lens of shipping. But at the same time, talking about her crush on the Doctor is unavoidable to some extent, because it does inform the choices she makes, and so on. Hopefully I've got the balance about right. (For the record: I think they could have been great together, but they met at entirely the wrong time in the Doctor's life for it to ever work out. Having declared my bias, I shall try to avoid employing it -- which should be relatively easy, as we're talking about Martha, not the Doctor -- but there it is and if you detect it feel free to ignore it.)
Credits: Many thanks to in_the_end for giving it a once over before I posted it; the pretty pictures are from Time and Space.
The starting point
In this section, I'm going to explain what I think the key points are about the Martha we met at the beginning of S3.
Firstly, and most importantly, she is not good at wanting things.
When Rose first met the Doctor, she had a sense that she wanted more out of life than she was currently getting, but no real idea how to get it. The Doctor unlocked the door to everything she could have dreamed of and a whole load more besides. Martha's problem is a step further back than that: despite the fact that she's got a good life on the surface of things, she doesn't actually know how to want things for herself, really.
She's used to sublimating her desires to those of her family, so used it to that it doesn't even occur to her to put her own views across. The initial sequence of phone calls that introduces her in Smith and Jones show her as the sensible one who has to broker deals between the rest of the family, and she seems to have a strong enough desire for everyone to get along all right that she doesn't want to add another set of competing demands into the mix. The language her family use is telling. It's all instructions: "you've got to ...", "tell ...", and they're all using her as a go-between -- none of them seem to talk directly to each other -- and while she tries to point out that some things are out of her power, she basically goes along with it. She's got a good sense of humour about it, but ultimately she puts others way ahead of herself, to the point I would argue of unhealthiness.
Apart from that, she's a huge dork. elliptic_eye wrote some excellent meta about her social dorkiness which can be found here, so I won't bang on about that at length. In short: she's not that socially adept, at least pre-YTNW. And she's a geek. This is one of those things that gets glossed as fanon a lot of the time, but I really don't think it is. The key evidence is in Shakespeare Code. Yes, there are lots of people much, much less geeky than any of you who've read this far down a DW characterisation meta post on LJ who have read Harry Potter and know basic ideas about time paradoxes. But it breaks the format of a non-wholly-comedy bit of SF to portray people who have more awareness than that (in much the same way as the Doctor is OMG amazingly clever but still takes ages to figure out the books->paper->trees connection with the Vashta Nerada), and so the fact that we're shown that awareness at all, rather than Rose or Donna levels of "explain the wibbly wobbly to me, Doctor", is evidence of her geekiness in my view. (Likewise, Mickey's little parallel universe speech in Rise of the Cybermen is nothing most pop-culture-aware people couldn't give, but he's portrayed as very geeky.)
As well as all this, of course, Martha has a number of general traits that make her good companion material: these are the things that make the Doctor go all googly-eyed at her throughout her first episode, and tend to lead some of us fans to run around the place telling anyone who'll listen (and some people who don't really want to) how she's made out of awesome, but they're also not things that change much over the course of her travels so I won't bang on about them at length after this. But in particular, she's observant (picking up on the Doctor's double heartbeat is an early example of this) and good at thinking on her feet in a crisis (the X-ray machine). She's clever, in the book-learning and the analytical senses (though she lacks confidence in this area -- that line about "if I can pass my exams" strikes me as an expression of a sincere doubt about whether she will, not just trying to seem not too arrogant; probably this is related to her family). She's compassionate and caring, though (the dorky thing again) not as good as some others at saying the right thing to people in their moments of need. And she can run well in impractical shoes.
Welcome aboard, Miss Jones
Martha makes a big decision at the end of Smith and Jones: not just to travel with the Doctor, but to get away from her family who are not only continuing their failtastic soap operatics, but also don't believe her about what happened on the Moon.
In the following series of adventures (Shakespeare Code, Gridlock and the Dalek two parter) Martha shows off her skills some more and quickly cements her companion bond with the Doctor (for all that she doesn't get a key until 42, they really sealed the deal with the handshake at the end of Smith and Jones, he compliments her repeatedly in tSC, his concern for her in Gridlock is not just of the "I got her into this" Five-n-Peri-in-Androzani variety; all the just-one-trip stuff was the Doctor and his issues IMO), while her initial flirty attraction turns into falling deeply for him.
OK, so this is a part where we have to talk about the shipping. There's a fundamental miscommunication between the Doctor and Martha at this stage that affects everything that happens after. The Doctor does go on about not being interested, but so does Martha. She's doing it as part of flirting with him, and she assumes that he's playing hard to get too. And he does like her a lot, the same way he does all his companions, and his boundaries are not those of humans, let's face it. It's mixed signals, at best. The key thing is: Martha wants the Doctor. Yes, she's having a great time travelling all of time and space, and yes, she's finding a sense of purpose and learning that there are corners of the universe that have bred terrible evils that must be fought, but she's suddenly discovered what it's like to have something that she wants, for herself. The Doctor has, inadvertently, taught her to be just a little bit selfish. Which is something that, really, she needed.
The Lazarus Experiment is a key turning point. It's the first time that Martha stands up to people who are important to her, tells them that what she wants is more important than what they want (leaving her family at the end of S&J doesn't really count, she just slopes off). Francine's shocked reaction is indicative of how unusual this is. But more importantly even than that is her standing up to the Doctor, refusing to go back with him unless he acknowledges that it's not yet another "just one trip". She wants him, but now she wants him on her terms. There's a parallel here with her departure at the end of Last of the Time Lords: the difference is that on this occasion what she's asking for (at least explicitly) is something he can give. He does so willingly, they hug, and go on to new adventures. In some ways, everything up to here has s all been one extended "welcome aboard".
As with Donna a season later, Martha really gets put through the wringer in the second half of S3. 42 sees her stranded in a near-death situation and coping with a frightening vulnerability from the Doctor, and then we're into a triple whammy of long stays in conditions ranging from nigglingly uncomfortable to literally postapocalyptic.
In a way, it's Martha's doormat-ish tendencies (which, while they may be receding as a result of her adventures, haven't magically disappeared overnight)
that help her get through the first two of these. The school in 1913 is a horrible environment for her to be in; she bears up with grace and humour, but there's a sense that she only achieves this poise because she knows it's temporary, that she sees it as an undercover mission. We get very little information on her feelings in 1969 -- there's always the possibility that her bit about working in a shop on the DVD is some sort of wibbly wobbly spontaneous generation of the time loop, rather than a reflection of her genuine emotions -- but it can't be a huge amount of fun, and unlike Human Nature there's the worrying possibility of being permanently stranded.
But of course the other thing that happens in 1913 is that a man who looks like and shares many traits with the object of her affections, and has stolen his body, falls for someone else. Naturally, she's jealous and upset, and really the fact that she does go on to empathise with Joan is a credit to her, although she only does this once she's reasserted her own confidence in the fact that she knows of a far bigger world than Joan with the bones of the hand moment. Martha is jealous -- it's natural for anyone, but Martha, still new to wanting things this badly, suffers particularly badly for it. And of course, that jealousy comes up again with her reaction to the Doctor and Jack obsessing over Rose at the beginning of Utopia, though I have to say the "oh, she was blonde" comes across a lot less bitchy than a lot of fandom paints it.
(The other big moment in Utopia that Martha gets raked over the coals for by some is persuading Yana to open the watch. I refuse to coutenance this: the rose-tinted picture the Doctor paints of Gallifrey since its destruction makes it perfectly understandable that she'd think any other Time Lord would be Good News. And he wouldn't even have had to mention the exist of his fellow rather less nice renegades to counter that impression, just tell the truth about his dealings with his people. I blame the Master most of all, the whole hiding thing was a deliberate plan when all's said and done; and the Doctor second. Though ironically I'd be prepared to bet Martha does blame herself.)
Anyway, then we've got the Year That Never Was, where Martha not only survives on the decimated Earth but undertakes her covert mission to unite it around a love of the Doctor -- a love of the Doctor she teaches everyone to share with her. (There are huge, huge problems with the whole thing on a plot logic level, but on a thematic one it is, for better or worse, the apex of RTD's vision of Who.) A lot of people claim that this constitutes Martha getting character development off screen, but I don't really buy it myself: what happens in the Year is what's been happening all season, on an epic widescreen scale. Yes, Martha gains new skills as a result of her experiences that we don't see her learning, but learning new skills only constitutes "character development" in RPGs.
And then, when it's all over, we're back at the Lazarus situation. Martha's love for the Doctor, amplified through humanity as a whole, has just saved the Earth, but at the same time she's come to realise that he doesn't love her back the way she wants to. Just as with Lazarus, she tells him she's leaving; this time, he can't give her what she wants, so she goes. There's also her family to consider. While going to look after them is pretty much an excuse, and she'd clearly stay on the TARDIS if the Doctor did turn round and say "Martha snoggings, yes pls", she does care about them deeply -- that's always clear even at the times when she's at odds with them -- and they've been through hell.
As is hinted very heavily in her Torchwood appearance, Martha gets recruited by UNIT on the Doctor's recommendation. It's fitting that she gets involved in saving the Earth; she's passed her exams and become a Doctor, which is frankly the best ending companions tend to get (they break down into four categories: death, random marriage, return home with nothing much really changed except some odd experiences, apply what you've learned and carry on the Doctor's legacy). The fact that she also becomes to a certain extent UNIT's conscience, the Doctor's representative within the organisation, as seen in the Sontaran two-parter, mirrors the way she was his earthly representative in TYTNW.
She's also found happiness with this timeline's version of the man she met at the end of the Year, Tom Milligan. This happens entirely off screen and so is an area of Martha's story that's got more fanon than canon, so I won't go on about it at length, but it seems to me that she's applied her new powers of going after what she wants, this time with a target who reciprocates.
We see in TW and S4 the culmination of Martha's development: the knowledge and skills, but more importantly the self-confidence and self-assurance, that she's gained through her time with the Doctor are all on display, without her having one hand metaphorically tied behind her back by her crush. Her relationship with the Doctor, while it may not be what she once wanted, is one that works for both of them, and is very much on her terms: she can call him at will, she gets taken home when she asks ...
And Martha is really the only one of the "main" new series companions to come out the other end of Journey's End intact. And she gets to bluff Davros along the way, not just standing up to him the way she did when she laughed in the Master's face at gunpoint, but actually threatening him -- that self-confidence again. She's got all the same key qualities she had back in Smith and Jones, but she's much more aware of them now.