He is incredibly easy really, with any character you decide to play as. Just wait until you have ALL the other quests in the area complete (except for the Mad Mel one). You don't even need to go turn them in, I never do. I just finish them all and then do Skagzilla.
It's quite easy to kill Skagzilla without being hit even once, especially with Lilith. Save your Phasewalk for when he's about to charge you and you can get out of the way easily. Keep moving and jump to the side whenever he tries diving on you, and occasionally shotgun his back legs when he lands if you can. Other than that, just keep moving and wait patiently for him to open his mouth and SMG the shit out of it before it closes (remembering to jump over/sidestep the beam) and he'll go down in maybe two beams.
Also shock weapons are horrendous against flesh/armour and launchers are practically useless against everything really (DAMN YOU GEARBOX). Just get a corrosive/fire/explosive SMG and go nuts. Any shotgun will work, even a neutral one providing it has enough power behind it. Hell, even sniping it works well. Just don't use shock or launchers. xD