why does Injun Joe kill the doctor?

Injun Joe kills him, but he doesn't seem to have planned the murder in advance. In fact, upon closer inspection it seems that all Joe really wanted to do was extort some money from the doctor.

THere is also some revenge thrown into this,

"Five years ago you drove me away from your father's kitchen one night, when I come to ask for something to eat, and you said I warn't there for any good; and when I swore I'd get even with you if it took a hundred years, your father had me jailed for a vagrant. Did you think I'd forget? The Injun blood ain't in me for nothing. And now I've got you, and you got to settle, you know!" (9.44-47)

As you can see, Injun Joe is out for two things in this case: money and revenge. That said, revenge or "settling" in this case seems to be a matter of five extra dollars, not a human life. Once the fighting starts Injun Joe is perfectly happy to kill Robinson, but it doesn't seem to have been in his original plans.