Attack his legs until his aggro meter kills up, after which he will start to fire rockets at you. Run towards P.A.X. to get the boss to damage himself with the rockets, and he will be in a vulnerable state for 9 seconds. Hack away at his health in that vulnerable state, then repeat the process.
P.A.X. weapon drops:
Normal Kill: P.A.X Imperator, Vital Boost v.3
Challenge Kill: P.A.X. Imperator V2.0
To complete the challenge, kill P.A.X. without using his own missiles against him.