Big Game Hunter is a quest given by Ernest in the Dahl Headlands when you first arrive. The quest involves finding a large piece of meat and placing it on a spit in a large caged area to lure out a massive skag called Skagzilla. You then must kill Skagzilla in order to win the quest.
I recommend standing right in front of the cave that Skagzilla comes out of at first, when he comes out he will roar for a few seconds. During this time shoot as many shots as you can straight into his mouth. This is where you score critical hits on skags and if you have a decent weapon (see below) you should be able to bring him down to atleast half health. Once you've done this, run in large circles and jump away from him whenever he leaps at you. Keep shooting towards his mouth until you've killed him.
Keep in mind you'll need to be using either Regular (No Elemental Damage) bullets and/or Explosive elemental bullets. Skagzilla is resilient to fire, corrosive and shock bullets. I highly recommend explosive as it seems to work better than no element at all.