The title of this achievement is a bit deceptive, because I found it only worked if the character you were controlling at the time defeats the Glasp. It's not a team effort, even if you collectively kill all 6 Glasps. For example, throwing a Smokescreen Bottle while controlling Moira but having the CPU Claire kill the Glasp will lock you out of the achievement. My strategy was to give all my bottles to Claire, and controlling her, I killed them all with either Explosive or Fire bottles. I found that Decoy bottles actually also work stupidly well against them.
As the achievement suggests, it's mandatory to play on Normal or Survival. On Easy, there are only 4 Glasps. There are a total of EXACTLY 6 (on Normal, anyways), and they come in groups of twos.
They are pretty easy to spot, as the screen will start to change when you are near one (the same as in Barry's chapters). The first two are on the same screen as the first workbench, as you make your descent when the tower starts to blow up. The next two are further on as you progress (after destroying the locks on the gate, but before entering the door for the outdoor descent). The last two are in the final part of the tower, after climbing down on the outside you will come back inside. You'll want to kill them quickly because it's a timed race to the bottom.
After you've killed all 6, the achievement will unlock. No need to finish the chapter and wait for the results screen for it to show up.