Equipped electric punch, winter shield, fire bird, and overkill. Overkill is optional. Since winter shield will give you invulnerability after leaving the skyline, you can substitute tunnel vision and take your time to go for headshots. All you do is jump back and forth off of the skyline for the entire fight. As soon as someone is in range pounce on them from the skyline. Winter shield invulnerability triggers whenever you pounce and will give you enough time to kill several more enemies without taking damage. The pounce attack counts as a melee attack, so you get the firebird effect plus the high damage of the attack plus electric punch 70% of the time. Watch the winter shield effect on your health bar. When it disappears immediately jump back to the skyline and pounce on someone else. Since there are skylines all around the fighting area, its always easy to simply stop shooting and jump on one whenever winter shield is about to run out. There's also usually an enemy nearby to jump on as soon as you hit the rail as well, so you get to keep your damage output high while remaining invulnerable for the majority of the fight. If you can do this right, you can take next to no damage during the whole fight while taking guys out left and right.
I would normally target songbird on stuff as soon as he was available, always looking toward the zepplins first, but if they weren't available whatever was in sight would do. I also threw out a few fully upgraded murder of crows vigors whenever I wasn't doing anything else. The nests they left behind kept guys from making it to the shooting area so fast when I was trolling around looking for someone in range to jump on. Other than that it was just rinse and repeat. Stay invulnerable as much as possible and make sure you don't go more than a couple of seconds without killing someone. It was actually pretty easy this way on hard.