I just happened across something I wrote back in '99 in response to a friend who'd asked, "How [do I] make a roleplaying character believable as a person (am I asking 'how to act'?)"
I thought my answer might be of some interest, though it's certainly not the only answer; so I'm reproducing it, lightly edited, here:
I'd say you're asking a combination of how to write and how to act. The first step is creating a believable character; the second step is portraying that character in a believable and consistent fashion.
Some assorted approaches that might help, in no particular order:
Think about the way the character talks. If they're highly educated, they may perhaps be inclined to use complex words and intricate sentence structures. But if they're a street kid, they're gonna talk like one. Diction and vocabulary are key here. You can even correct yourself: "She says, 'Hey, bring that flashlight over so I can examine the body.' No, wait, she wouldn't say 'examine.' 'Yo! Shine the light over here, dude!'" I think of it as putting a filter on my vocabulary and range of diction styles--limiting the character's vocabulary, for instance, to a subset of my own. (It's hard to play a character who's smarter than you are unless you get help from the GM and/or the game mechanics.) Don't caricature speech styles unless you're playing a heavily-genred game; if you're playing Champions or Space:1889 it's fine to do a stereotyped Brooklyn or Cockney accent, but in a more serious game with less broadly-drawn characters, think carefully about the way the character would really express themself.
If you're a more visual person than I am, think about the way the character dresses. This can tell you (and others) a lot about the character if you know how to interpret such clues. Not something I'm good at.
Think about philosophical questions like "What are the character's goals?" and "What do they want?" and "What intangible ideas [such as truth, honor, greed, loyalty, . . .] are important to them?" Even if the character wouldn't think about such things, you should do so on their behalf. (And if the character wouldn't think about such things, that's useful to know too.)
Delany gave us a character exercise at Clarion that involved distinguishing between habitual actions (those performed often/repeatedly); purposeful actions (to accomplish something); and gratuitous actions (the sorts of little odd unnecessary things real people do). Think of three of each type of action for your character. Visualize those actions clearly and specifically. Use some or all of those actions when playing the character in the game.
Delany also told us "Character is not in the details but in the tension between the details" and "Specific observation points give characters greater intensity."
Give the character hobbies and interests that are totally unrelated to the plot.
Give the character problems and disadvantages--minor physical disabilities, phobias, quirks. (Talk these over with the GM first to avoid having, for instance, a character who's too out of shape to even walk fast, if the game will require the character to do a lot of running. Disadvantages should be interesting without disrupting the game.) I tend to overdo this, putting such limits on my characters that they become ineffective in combat and such; Rob and Jamie and Arthur have a running joke that my characters in Hero-system games get an automatic 15-point disadvantage called "Played By Jed."
Finally, try to achieve emotional identification with the character. Try to imagine yourself to be the character, to the point that you experience the emotions the character would experience. Okay, okay, there are different schools of performance; some actors would say it's silly to get that far into character. But for me, being that far into a character is the main point of RPGs.