All the answers above are correct, and I will say that it is a common phrase in the United States. In some circumstances "chum up" implies that you have an ulterior motive, as in you want something from the person you're "chumming up to."

"To chum up" means to become friendly or form alliances. "Chum" = friend.

It is not a common phrase at all, but it would mean to make friends with someone.