There are a few tactics I like to use for the City. For a fairly general build, I like to go in with at least a Tac Shotty and a shell box (ideally). I use that combo against isolated pain elementals, then soften up the Agony Elemental with a few BFG blasts. Once I've critically injured the AE, I switch back to the Tac Shotty or melee and clean up. Finishing off the AE with a BFG usually destroys the skulls, which negates the larger point of why I go in there (if I make it so far, I NEVER skip the Halls of Carnage). Of course, things seldom go that smoothly, and if I wind up surrounded (as can happen right after pulling the lever), I REALLY like to have a Phase or 2 on hand.
MAD and MBM are both friggn' hilarious on this level (MVm has fallen out of my favor as of late, maybe I'll give it a go later). MAc is good as always, but you need to be careful to avoid getting swarmed- focusing fire on pain elementals first is the best way to control lost soul spawning. With a twitchy-ass trigger finger, you can generally mop up any straggling souls once the elemental is dead.
My "default" build is what I call Malicious Rocket Scientist- MMB followed by Juggler and Whizkid. I try to build a Tac RL in favor of the ML, so I can focus on rocketjumping to get in some poor bastards' faces with a modded Piercing Knife. It works pretty well here, since it lets me mix up my engagement range and permits me use all the tactics described in this post and earlier in this thread. (offtopic pssst- rocketjumping is stupid good on CC+)