Ajax is a Crimson Lance Commander, and presumably the higher in command of the Lieutenants Kyros and Typhon. Ajax will not spawn in game until the player has been given the side mission This Bitch is Payback, pt. 2. This mission requires first completing This Bitch is Payback, where the player must kill both the Lieutenants.
Ajax has a massive shield, and (relatively) low health. While standard corrosive weapons do work against Ajax, a high quality Shock weapon (like the S&S Orion) is enough to 3-shot him.
Devastators will appear during the fight, as well as many other Rocketeers. It is best to take out the other Lance troops first, while avoiding Ajax himself.
If approaching from the east, a Monster can be used to great effect, lobbing missiles through the gap. Don't lock on, and aim low, or the missiles will probably impact the rock.
There are a few places that can be used to lure Ajax out of his area:
Approaching from the nearest Catch-A-Ride, there is a cliff on the right that can be jumped on - head left from there.