We all know that Injun Joe is one bad dude. The question is, why is he so bad? Why does he do the awful things he does? If we're talking strictly about motivation, then it's safe to say that there are two forces at work: greed and revenge. Take the case of Dr. Robinson. Yes, 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:
Potter took out a large spring-knife and cut off the dangling end of the rope and then said:
"Now the cussed thing's ready, Sawbones, and you'll just out with another five, or here she stays."
"That's the talk!" said Injun Joe.
"Look here, what does this mean?" said the doctor. "You required your pay in advance, and I've paid you."
"Yes, and you done more than that," said Injun Joe, approaching the doctor, who was now standing. "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.
Revenge also motivates Injun Joe's unsuccessful attempt at attacking the Widow Douglas. As he explains to his companion:
But her husband was rough on me – many times he was rough on me – and mainly he was the justice of the peace that jugged me for a vagrant. And that ain't all. It ain't a millionth part of it! He had me horsewhipped! – horsewhipped in front of the jail, like a nigger! – with all the town looking on! Horsewhipped! – do you understand? (29.30)
Joe admits to being treated poorly by the Judge many times, but his biggest grievance with the now deceased Mr. Douglas had something to do with a vagrancy charge, perhaps the very same one Robinson's father got him put in jail for. Injun Joe certainly does seem to have a monkey on his back – it's his inability to let go of his grudge with the Widow that stops him from splitting for Texas.
Injun Joe also seems to have some issues regarding race. Since he's part Native American, he is looked down upon in society. When Huck tells the Welshman that the "deef and dumb Spaniard" is actually Injun Joe, the Welshman thinks he should have known the Spaniard was Injun Joe because, in his opinion, only an Indian would do such horrible stuff to a woman. This is certainly an offensive stereotype, but Injun Joe doesn't seem to mind it. In fact, he even seems to like it: when talking to Dr. Robinson, he uses his "Injun blood" as an excuse to justify his aggression. At the same time, he can't stand being treated like a second-rate citizen; he is conscious of a racial division, and he doesn't want to be treated as an inferior person.
Now, this isn't meant to excuse Injun Joe or make him a sympathetic character. He really is a cold-blooded killer and a thief. But he shouldn't simply be written off as a one-dimensional "bad guy." He's a complex character, and he's worthy of analysis.