Tools of Characterization
Characterization in A Beautiful Mind
Speech
We can tell that John's frenemy Hansen is kind of an arrogant prat by the way he talks to John when they first meet each other:
HANSEN: I'll take another.
JOHN: Excuse me?
HANSEN: A thousand pardons. I simply assumed you were the waiter.
Er, yeah, no you didn't, Hansen—you're just trying to psych out your big competition in grad school.
Of course, we can also tell that John isn't a pussycat from his response to Hansen:
JOHN: Well, Martin Hansen. It is Martin, isn't it?
HANSEN: Why, yes, John, it is.
JOHN: I imagine you're getting quite used to miscalculation. I've read your pre-prints, both of them, one on Nazi ciphers, and the other one on non-linear equations, and I am supremely confident that there is not a single seminal or innovative idea in either one of them. Enjoy your punch.
So, yeah, as you can see here, dialogue really shows off these dudes' true colors from the very start of the movie.
Actions
Well, nothing drives home the severity of John's illness like his behavior in the mental hospital. John takes a while to come to terms with the fact that he's been having hallucinations, and he even carves up one of his arms looking for a "tracker" that he believes Parcher had injected there.
That shows you just how desperate he is at that moment to prove that he's been living in reality. Which, of course, he hasn't.
Family Life
John's life with Alicia definitely shows us a softer side of our favorite genius. For example, once he and Alicia are married and she's pregnant with their son, he decides it's time to stop putting himself at risk through his "work" with the Department of Defense. His work has always come first, so it's pretty striking that he's willing to give up the spy games part of it for Alicia and the baby.
(Of course, the spy games are all imaginary, so he's not really giving anything—but he doesn't know that, so it's still a meaningful sacrifice.)