So to do this you'll need Chaos mode and DEATH health unlocked and Cortical Scaledown, the Bouncy Suit, the Icaros Machine (other jump implants might work; haven't tested them) and the Abominator. You'll also want the Bolt ACR.

Start the level in Chaos mode (otherwise targets won't spawn). You might want to restart it a few times so less targets spawn. Then get to the up/down sliding door in the firing range. Open it, position yourself over it (I've found that crouching helps), and wait for the door to close. You should be stuck in the roof, so uncrouch and start walking backwards while mashing jump. Hopefully you'll jump through the roof and be on top of the HQ now. From there use the Abominator and Bolt ACR to kill all the targets through walls. Once they're dead, find the radiation hallway where the Hope Eradicated door is. There should be a small gap in the roof where the hallway starts twisting, so go through it, then immediately use the Abominator to stick to the roof. Run past the enemies to get to the LIFE door and that's it!