Mark Twain wove modern social implications into his stories. In The Adventures of Huckleberry Finn, Twain points out racial injustice towards blacks in the South. In the person of Miss Watson, he symbolizes changing attitudes of the time: she frees Jim in her will, guilty for how she treated him. Huck believes society when it teaches that blacks are an inferior race, but he cannot agree and says he'll go to hell for it. In Chapter 31, Huck writes a letter to turn Jim in, but then says...
“All right, then, I'll go to hell”—and tore it up.
...Jim cares for Huck like his own family and Huck loves Jim despite what society "preaches."
Twain tells us how he feels in his stories—Injun Joe may represent the fallacies and/or fears that some whites have of Native Americans then. Injun Joe is blood-thirsty murderer. Where in Huck Finn Twain shows Jim in a positive light to demonstrate that he is a good man first and a black man second, this is NOT the case with Injun Joe.
Twain does not address social injustice against Injun Joe. Neither does the straightforward Twain presents Injun Joe as a victim of racism, OR maintain that Injun Joe is representative of all Native Americans. Injun Joe represents someone outside the boundaries of polite society: just as Twain did with the Duke and the King in Huck Finn, he presents people who are marginalized—outcasts of that society at that time. Twain is not saying that a "Native American not held on a reservation is dangerous."
Injun Joe happens to be Native American—perhaps Twain uses him because in his time, Joe's race is still an unknown quantity. In Huck Finn, the villains are white men: Pap, the Duke and the King, the Shepherdsons and the Grangerfords, etc.
In The Adventures of Tom Sawyer, Injun Joe as a villain—not the only one (others are white); he's just the worst.
Injun Joe is...
...an angry, vengeful, amoral man.
...his identity is so closely tied to his being a Native American that the townspeople—and the narrator—cannot think of him except in terms of his being an Indian.
Twain represents Injun Joe based on how he is perceived by society: they can no more separate Joe from being a Native American than the whites in Huck Finn could make a distinction between Jim, the man, and Jim, the black slave. The difference is that Jim gave no offence: Injun Joe does.
We see Injun Joe carry out his vengeance; he tells the Doctor:
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...
He kills the Doctor and blames Muff Potter.
Injun Joe is a murderer and is portrayed as the worst kind of man: vicious and evil. Killing is not the only evil he perpetrates...he doesn't kill women:
You slit her nostrils—you notch her ears like a sow!”
Joe is a sociopath..."embodying the possibilities of human evil." It is not a surprise that Injun Joe's tries to kill Tom and Becky in the cave.
Joe is a Native American, but Twain is not trying to vilify Native Americans: Twain successfully uses these details to make the man—Joe...who happens to be a Native American—a truly frightening character.
Twain shows that blacks are not inferior: they are men. Native Americans aren't evil as a race: but a man can be evil despite his race.