Tombstone should be played as an urban legend-grade terror. He likes violence, he loves to be viewed as an extremely dangerous man (which he is), and he has a degenerate sense of humor. Consider him an embodiment of the worst the underworld has to offer : strong, professional, unstoppable and sadistic.
He likes to play with his preys, and is a bully. Being stalked by Tombstone is an harrowing and quite possibly fatal experience, even for characters of consequent power.