Besides being a professional swindler, counterfeiter, and general law-breaker, Compeyson is the man who jilted Miss Havisham at the altar all of those many years ago. He was in cahoots with Arthur Havisham all along, and eventually he linked up with Magwitch.
When Magwitch and Compeyson were caught forging signatures, Compeyson got off easily with only half the jail sentence that Magwitch received. Why? Because of his "curly hair and his black clothes and his white pocket-handkercher" (42.32). The court appreciated Compeyson's gentlemanly ways, and, thus, went easier on him. Compeyson and Magwitch hate each other and are forever fighting. Eventually Compeyson drowns after fighting Magwitch in the Thames. Did Magwitch kill him? He's not saying.