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.