The Black Cat is kind of a snot. There, we said it. We love him, but he really does have an attitude problem.
"We could be friends, you know," said Coraline.
"We could be a rare specimen of an exotic breed of African dancing elephants," said the cat. "But we're not. At least," it added cattily, after darting a brief look at Coraline, "I'm not." (4.19-20)
Over time, though, the cat learns to respect Coraline; he even teams up with her and helps her escape from the other world. In fact, he even catches one of the children's souls from a runaway rat!
The cat may act superior and be very brave, but even he can have his moments of fear: "[Coraline] put down a hand to stroke it, and could feel how hard its heart was beating. It was trembling like a dead leaf in a storm (10.57)." Don't call him a scaredy cat though: he has good reason for being afraid, and he pulls himself together when it counts. Plus, he could probably eat you.
In the end, the Black Cat really becomes Coraline's friend and companion. He moves in with Coraline once they return to the real world, and we're willing to bet that the Black Cat will keep looking after Coraline in the future.