What were some examples of criminal behavior in the culture? How did they control crime?

In Things Fall Apart, the Igbo tribe integrates their judicial system into their culture; it is not a stand-alone institution like it is Western civilizations. The tribal elders regulate crime using the death penalty, exile, and reparations.

Okonkwo is guilty of three "murders": Ikemefuna, Ezeudu's son, and the British messenger. The first killing is intentional, but since it was sanctioned by Chielo, the Priestess of the Oracle of the Caves, no penalty was administered. The accidental killing of Ezeudu's son resulted in Okonkwo's exile for seven years. This was considered by the tribe to be the feminine version of murder (accidental, unintentional). So says Enotes:

Okonkwo’s immoral actions affect the community. During the funeral rite for the elder Ezeudu, Okonkwo’s gun accidentally explodes, killing Ezeudu’s son. Okonkwo’s crimes enrage the Earth Goddess Ani, for he has consciously and unconsciously chosen death by beating his wife, killing Ikemefuna, and now, killing Ezeudu’s son. His irrational actions are destroying the moral fabric of traditional life. Therefore, Ani banishes Okonkwo to Mbanta, his mother’s village, for seven years.

The third murder, the killing of the British messenger, might well have been the "male" version of murder and, therefore, punishable by death by the tribe, but the British judicial system had taken over at that point and, of course, Okonkwo commits suicide. This last killing (of the self) is the greatest abomination of the tribe, and Okonkwo's body would have been left unburied:

It is against our custom...it is an abomination for a man to take his own life. It is an offense against the Earth, and a man who commits it will not be buried by his clansmen. His body is evil, and only strangers may touch it.