After Skagzilla is defeated, he will simply be in his pen when he respawns rather than emerge from his cave.
A Skagzilla roar can be heard when his cave is approached after he is slain, but no accompanying giant skag to go with the noise.
An easy method to farm Skagzilla after he is killed the first time is to jump into the pen and lead him to the exit. He cannot follow through the small gate, and will instead repeatedly roar in frustration, allowing for easy critical hits. This should prove easy, as Skagzilla's attacks can be easily sidestepped if another player grabs his attention.
This method can also be used to kill him for the first time if a character dies after Skagzilla has already appeared. Jump to the exit platform on the far south end of the arena, fire at him to attract his attention, and then fall (or get pushed) onto the ground. He will rush to the gate and repeatedly roar, providing an easy kill, though this necessitates walk or a grenade jump back into the arena.
It's possible to grenade jump up to the cover right above the entrance. Doing so makes Skagzilla roar in frustration.