How do I write a speech about: "The battles we fight within ourselves are the ones which ultimately define who we are" in relation to The Crucible?

The best example of this can be found through the character of John Proctor, so if you are looking for a model to use for examples, he's a great one to go with.  In the beginning of the play, Miller explains that

"the steady manner [John] displays does not spring from an untroubled soul."

This indicates right off the bat that John is someone who has an inner conflict, a battle within himself, so that fits right in with your thesis.  His battle springs from the fact that he committed an affair with Abigail, and feels like a total and complete sinner and hypocrite as a result of it.  He does not feel like a good man at all.  He battles with that feeling throughout the entire play.  It bothers him when his wife wants him to go to Salem and reveal Abby for a liar; he knows that he too has been a sinner, and doesn't feel like the man for the job.  However, despite his feelings of unworthiness, he goes into town, and tries desparately to free his wife and friends from the paranoid graps of the courts.  This shows his character; he is a man who will do what is right, even if it is hard for him, uncomfortable for him, and, in the end, if it will reveal him for his sins.  He confesses to adultery right there in the courts, because he knows it is the right thing to do.  He battled that confession for a long time, but he is a man of integrity, so did the right thing.

At the end of the play, after he finally rescinds his false confession and is heading to the gallows, he feels like he is, for the first time in a long time, that "I do think I can see some shred of goodness in John Proctor."  He feels like a good man, finally. He has resolved that battle within himself, and through fighting for what was right, and sacrificing his life, he feels like he has paid penance for that battle.

For a counter-example, look at Abby.  The battle she was fighting within herself was her desire to be with John, versus societal codes that forbade them to be together.  In her life, her selfish desire to be with him won that battle, and caused the death of many people as a result, including John himself.  Her inward battle revealed her to be a conniving, selfish girl who would go to great lengths to get her way.

I hope that those thoughts helped; good luck!