Qarun was one of the people of Moses –peace be upon him-. It means that he came from the society of whom Prophet Moses was sent to.

Qarun was not a family of Moses, because family of Moses only consisted of faithful men and women, courtesy of the enjoin and treatise he brought. Because the title family doesn’t embedded to unfaithful kins.
It is likely that Qarun was, at the beginning, believing in the enjoin of Prophet Moses. But when Allah the Glorified and Exalted opened for him path to obtain wealth and His treasure, he forgot the enjoin that he was called into because he was busy taking care of his wealth, and then he acted Despotically towards the truth and good.