The first section of this chapter is mostly story-driven, with a lot of walking. Walk forward, and a ramp will lower for you ahead. Cross it. Keep walking down the long hallway and hit the switch on the elevator at the end, which will activate as soon as all three of you are on it. Ride the slow elevator upwards, getting a glimpse of Titan Invictus. Turn around and keep walking until you hit the next checkpoint.
Keep walking forward and the hit door switch. Walk through it and your squadmates will hit the switch on the left side. After a brief conversation, go down the staircase to the right of the door you came in, turn left, then turn left again to find a Servo Skull (Workshift 465 2/5) and pick it up. Head back upstairs and turn left. Hit the switch on your left to activate a downward lift. When you're all the way down, hit the door switch in front of you.
Walk through the door, and Leandros will use the panel to contact Sgt. Midas. After that, the door switch on your right will turn on, so hit it. Keep walking. Notice a pattern yet?
Follow the linear walkway for a bit, and then you'll spot some Orks trying to infiltrate the Manufactorium through the pipes. Take them out and keep moving forward, following the objective marker once it's all clear. Just before you reach the objective marker, there should be a fork in the road. Take the left path to collect another Servo Skull (Inquisitor Drogan 2/5). Go through the door on the right.
Hit the switch to lower the platform on your right and run across it to the next door.
Go through the door and approach Inquisitor Dogan to trigger a cutscene.
Video (NOTE: Chapter 6 starts at 0:47):
Go through the door on your left with Drogan and you should receive a new objective (Get to the Generator Room). Walk forward to activate a cutscene. If you're prepping yourself for a bossfight with Grimskull, you'll have to wait a bit. At the moment he'll just send a bunch of flunkies at you, almost all of them ground forces, and you can deal with them easily. Once everyone's dead, follow the objective marker to the door on your far left.
Once you go through the door, a short conversation with Drogan, concerning Grimskull, will trigger, then you can move forward through the next door. Your old friend the Support Pod is waiting for you there, and swap out your Bolter for the Pod's Plasma Pistol. Use the lift on your left to move up and trigger your first fight with 'Ard Boyz, who are summoned by a Weirdboy. Kill them with your Plasma Pistol and keep moving towards the objective marker. Take the lift down. Step out of the lift to activate yet another checkpoint.
Move forward and open the door. Keep moving forward and open another door into the Generator Room. Objective Complete!
Move forward again, picking up some ammo to your right. Once you've dropped down to a lower level, Drogan will open a chamber for you. Go inside to trigger another arena fight. Grimskull will taunt you from an upper floor, while the Weirdboy from earlier will summon flunkies to attack you. Kill everyone. About halfway through the fight, another checkpoint will activate.
Inquisitor Drogan asks you to activate the Servitor station, but you still have 'Ard Boyz to take care of. When you get bored of them, go and activate the panel to trigger a short cutscene where Grimskull will try to grab the power source for himself, but fail. You need to put the power source back in the canister, which you can do by walking up to the canister and pressing //. This triggers a short cutscene and ends the chapter.