Christmas leaves this repressive family because they will not nurture his search for identity. He longs for companionship, sex, and fun – a normal teenage life that he cannot get in this household. He also wants to come to some sense of harmony with himself and to resolve his racial identity issues.
Christmas experiments with his racial identity during this time, alternating between passing as white and black depending on which woman he's sleeping with and which town he's in at the time. He's almost trying on different identities, trying to see which one fits.
Miss Burden's attempts to "reform" Christmas by turning him into a devout black activist only alienate and confuse him further. While he's uncomfortable in white society, he seems totally unwilling to give up his position "passing" as a white man, and the idea of helping black people seems disgusting to him. This inability to reconcile his simultaneous desire for and hatred of both whiteness and blackness leads him to want to kill Joanna.
If Christmas's goal in life was to find himself, and to discover a place where he could live in peace, then the murder he commits actually gets him the exact opposite of what he wants. Rather than finding a place to settle, Christmas is now doomed to repeat what he has done his entire life – run away.
Christmas's literal destruction is tragic because it's an act of racial hatred, ignorance, and fear. But it's also tragic since he has never actually realized anything about his racial identity. He has spent his entire life shutting people out and refusing to bond with anyone. He dies socially isolated and clueless about himself, one of the greatest tragedies possible.