Why does the man want to kill his dog?

The man wants to kill the dog to warm his hands, which will enable him to build another fire.

The sight of the dog put a wild idea into his head. He remembered the tale of the man, caught in a blizzard, who killed a steer and crawled inside the carcass, and so was saved. He would kill the dog and bury his hands in the warm body until the numbness went out of them. Then he could build another fire.