This chapter starts with Booker assisting the Vox in breaching the factory gate. Help the Vox defeat the enemies (regular goons and a motorized patriot) until you receive an objective update requiring you to destroy the Zeppelin. Get on the sky-line and follow it until you can leap off onto the ship. Once aboard, fight your way to the engine room, destroy the engine then jump off to safety.
Before leaving however, turn around and jump on the partition between the two long staircases to find a telescope some ways behind the giant Fink statue. Battle past the courtyard to enter the factory proper. Look for a toppled cart of sorts on the ground here for a piece of Gear. Check the small room to the left of the elevator to find another Gear behind the Dollar Bill vending machine.
The Vox soldiers will now attack Booker from the moment you step off the elevator. Kill them then check the table below the massive clock on the right for a Voxophone. Also beside a corpse on the floor is the Undertow Vigor. Loot the nearby desks before heading through the door.
The enemy horde consists of regular goons (both Columbia guards and Vox Populi), Fireman and a Handyman. This is the perfect place to try for the Hazard Pay achievement. Locate then hide behind the Tesla coil after asking Elizabeth to tear it in. Stay there to lure the enemies so that they get zapped to death when they approach. Kill everyone; grab the Handyman's random Gear.
When the area is finally clear, explore a bit more. Just past the Tesla coil is a telescope by the railing that you can use. Once you're ready, boost Elizabeth up and distract Fitzroy. The story sequence that follows grants access to a couple rooms that cannot be reached before.
After Elizabeth leaves for the room ahead, turn to Booker's left for an Infusion phial then go other way and pick up the Voxophone next to Jeremiah Fink's remains. Step onto the airship, knock on the door then set a course for the ship. Working Class Hero achievement / trophy unlocked.