Dain kills Azog on the inside steps of Moria (I think right at Gate 1) as Azog is trying to escape from the last battle of the Great Dwarf-Orc war on the steps of Moria. When Dain comes out of Moria, Thorin and Thrain want to continue the battle and re-take Moria. Dain disuades them by saying the Dwarven army is too weak to accomplish that task and that he felt a greater evil in the Mines than even the orcs. Presumably, this is an allusion to the Balrog that goes over the cliff with Gandalf in FotR.
Bolg is Azog's son.
My guess is that Azog is being used to give some form of on-screen villian that people will cheer for when killed by Beorn - I hope. I think jumping through the fire to go after the orc that cut off your grandfather's head, carved "AZOG" on his forehead, hacked his body up and fed it to carrion as a pretty decent reason. Given the situation, it looked like they were dead dwarves walking, might as well take out the bane of your family if you can.