The first chance you’ll get to acquire this is in the latter half of Chapter 5, just before you enter the Unitoligist ‘cathedral’. Obviously you will need to purchase the Line Gun (9000 credits) before attempting this and although not strictly necessary, a full Stasis bar is definitely recommended. It will make the achievement MUCH easier. With that in mind, I will start at the last opportunity you will have to make purchases.
About midway through Chapter 5, right after you deactivate the Gravity Core, you’ll find yourself in a small room with a Shop that is separated from the main room by a small hallway. The main room contains a Workbench, and a Save Station and duct entrance concealed by three ‘pillars’ with Kinesis Grips, suspended from the ceiling via a rail. Use Kinesis (LT+B) to move the ‘pillars’ out of the way to expose the Save Station.
Unless you plan on keeping the Line Gun, I would recommend that you make a save before purchasing it. You will need at least 9000 credits on-hand (or sufficient ammo if you have already have it), alternatively you could just sell something. A newly purchased Line Gun comes with 5 rounds, which is ample for the achievement and it is not necessary to upgrade it in any way. Optionally you could purchase some Stasis Refills, just in case you run out or you can’t be bothered to wait for it to recharge :)
Enter the air duct via the entrance to the left of the Save Station.
Make your way through the duct, turning left and left again, before the ceiling gives out and you fall into a small storage room. Behind you you’ll find 10000 credits and 4 random pieces of ammo. Enter the next room through the door at the end of the hallway in front of you on the left.
In this room you will find the Line Racks Schematic behind the pallet of crates in front of you (as you enter). To your left, on a crate is a container with credits or ammo in it and on the opposite wall you’ll find 2 random pieces of ammo on some shelves. Exit through the door on the left into a room with red flooring.
There are two Lurkers hidden in this room. Turn right and head down the hall to the next turn and look to your right down the next hallway. DO NOT step out of the light yet. At the end of the hallway you will see a slope that leads up to a vent and light hanging from the ceiling above. This is the vent they will come from.
Make sure you have the Line Gun selected and sufficient Stasis before triggering the Lurkers. Don’t go too far down the hallway or you will inadvertently trigger a Slasher. Now the easiest way to sever all three tentacles with a single shot is for the Lurker to be standing on a horizontal surface, but when they first emerge from the vent they’ll jump on to the wall on the left (their right) in a sideways position. They will eventually jump down, but they will be shooting at you the whole time. To avoid getting hit head back to the door you entered through and turn back and face where you came from. When they come around the corner they’ll both be on the floor. Use Stasis to slow them down and aim (LT) for the middle tentacle (as close to their body/head as you can) and let it rip (LT+RT). If done correctly the achievement will pop right away. Don’t worry if you mess up, restarting the check point will put you back in the small storeroom you fell into from the air duct.
If you plan to keep the Line Gun, remember to watch out for the Slasher that will attack you from the right as you go up the slope.
Hope this helps :)
Video courtesy of PowerPyx