Jim is suspected of having killed Huck Finn. However, Jim turns out to be a character who deeply cares for Huck's well-being. When Huck and Jim first meet on Jackson Island (and Huck explains the story of how he got there), Jim explains:

I 'uz powerful sorry you's killed Huck, but I ain't no mo' now.

From the beginning of the novel, Jim shows kindness and compassion toward Huck. He is his friend, and he grows closer to him throughout the story.

In chapter 11, Huck dresses like a girl (calling himself "Sarah Williams") and goes into town from Jackson Island to find out news. The woman (Mrs. Judith Loftus) begins to tell Huck stories about "Huck's death" (which never really happened), and she explains that some people think that his Pap (his father) killed him:

Some think old Finn done it himself. . . . Most everybody thought it at first. He'll never know how nigh he come to getting lynched. But before night they changed around and judged it was done by a runaway nigger named Jim.

This information, that Jim is now suspected of killing him, alarms Huck, and he almost gives away that he knows more information than a stranger in town should. However, he stops himself from speaking and listens to her story.

She explains that they suspect Jim is guilty of the crime because he ran off the same night that Huck was supposedly killed. There is a $300 reward for anyone who turns Jim in; there is a $200 reward for Pap. This reward is biased and unfair. There is little to no evidence (other than timing) that Jim harmed Huck. However, the reward for turning him in is more (probably because of his skin color and status as a slave) than it is for Huck's white dad, who was known to be a drunk and an abusive father.

In a larger sense Jim was accused of having the audacity to think a slave was a real man.  Only Huck figures out that Jim is more than property in his moment of crisis when he decides he will go to hell rather than turn Jim in, his conscious, like a yellow dog has poisoned him, Huck says.

Other than running away (which Jim decided to do after hearing that he might get sold down the river), Jim didn't actually do anything wrong. However, everyone in town thought Huck was dead. At first, Huck's Pap was the main suspect, because everyone knew Pap abused Huck. But when Jim disappeared the morning after Huck's "murder," the townsfolk deduced that it must have been Jim who killed Huck. According to the townspeople, in addition to being black and a slave, the fact that Jim ran away made him look guilty.

Jim was accused of killing Huck. The town's people managed to narrow the list of suspects down to Jim or Huck's Pap, and most chose to accuse Jim because he was black. This is why Jim ran away to Jackson's Island where Huck eventually met up with him.