How does Miller convey the personalities of the characters in The Crucible, particularly through dialogue?

Because this play is set in the late seventeenth century, the manner of speaking is already very different from modern and contemporary speech. In order to be able to detect character differences in speech patterns and word choice, one must read/ listen very carefully. The original version of the play text contains many additional stage directions and expository remarks by playwright Arthur Miller, and this material provides additional insights.

If one is looking mainly at dialogue to determine personality, then one has to look at word choices and the behaviors conveyed through them. John Proctor is well-spoken and intelligent, but his emotions are often revealed in his word choice; he also tends to curse, which reveals a man who does not let religious piety affect his determination to speak his mind.

Abigail is an example of a character who manipulates others with her words and behavior. She uses language to cajole and threaten, and is often flirtatious and charismatic (which explains why she gets others to do her bidding). She does this in the first act with the other girls when she thinks they might reveal the things she has done in the woods. Her dramatic words stun the girls into obeying her:  

And mark this—let either of you breathe a word, or the edge of a word about the other things, and I will come to you in the black of some terrible night and I will bring a pointy reckoning that will shudder you. And you know I can do it. I can make you wish you had never seen the sun go down! 

We also see in this speech and others spoken by Abigail that she is arrogant and has a sense of self-importance, and this behavior seems to affect the other girls, in particular Mary Warren, when they are asked to become officials of the court. Seeing Abigail so confident and bold convinces the other girls they need to act as she does, and in this we see the roots of the mass hysteria conveyed in the girls' playacting in the courtroom. John Proctor become angry at Mary when she continues to mention needing to be in the courtroom when he thinks she should be attending to her duties as his servant. He sees through their pretense, but the court officials choose not to. As he grows more frustrated with the situation, his dialogue becomes more intense and graphic, as when he desperately tries to convince the court that Abigail is out for vengeance in accusing his wife:

She thinks to dance with me on my wife‘s grave! And well she might!—for I thought of her softly, God help me, I lusted, and there is a promise in such sweat! But it is a whore‘s vengeance, and you must see it; I set myself entirely in your hands, I know you must see it now. My wife is innocent, except she know a whore when she see one.