Krom is the second main story-line boss within the game. By main, the bosses that lead you to gathering a key to The Vault and furthering the storyline. This is the first key your character finds without the help of the mysterious woman assisting you much and directing you through it. Krom, as described in quest text quite sadistic and an overall nice guy (or as the Sledge fight goes, "P.S., you're not friends."
Krom has a large turret at the tops of the garrison in the instanced quest area. When starting out, it is recommended that people stick to the upper levels as much as possible while navigating through cover and fighting through small camps of Bandits. Once you reach one level below him, he will begin to fire on you. Do not take this lightly, he WILL kill you in a matter of 3 to 5 seconds. Take cover quickly while fighting.
This is when it begins to get tricky, equip your strongest shield possible (if you're Lilith use Phasewalk.) Move up the RIGHT pathway (which should be the default one if you followed the path right) and just sprint up it, ignore enemy fire. When you reach the top, pause and crouch, staying behind cover. You can see and shoot at Krom, but he can't shoot at you if you position right (You have a nice margin for error here, but not too much). Edge out a bit and clear out other enemies, then simply snipe Krom with all you got.
If you want a more edgy way to do it. Do the same way, and when you clear out the enemies up top simply run to the right, and jump over his stream of bullets and run in the opposite direction, playing chicken with his gunfire. When you defeat him, you get a Repeater that does significant electrical damage.