This one can be a bit glitchy (after reading around a bit it seems this was the same when it first came out on PC last year).
Essentially you need to complete a level using ONLY the Tarantula turrets. You unlock these when you get the Tech-Adept to level 5, but you cannot actually equip them until you get onboard Van Wynters ship - so follow the story until that happens and you're shown the Construct menu.
Now make sure you swap the existing two Constructs for the Tarantula guns in each slot - you can equip different gear etc here too if you've got the right passive upgrades. Also make sure your own character does NOT have any gear equipped that reflects damage or auto damages enemies (like digi-weapons) as they can void it too.
Now the fun part - completing a level and it actually counting. I had to do 10+ levels, doing exactly the same thing each time before this unlocked.
Firstly you need to complete a level AT OR ABOVE YOUR OWN. It seems that if you level up during a mission and that takes you over the starting level rating then it won't count. So try and do a level that is at least one level higher than your own (you can change the level difficulty from the menu before you start). I would recommend doing this on a random mission rather than a story one as a lot of the story levels see you getting other characters that help you, or finding Guardsmen that fight with you, or hacking turrets to assist etc - none of these levels will let you get the achievement as the game seems to see it as the kills being by something other than your turrets.
Once on a level do not attack ANY enemies directly yourself with either guns or melee - just don't fire at all (some levels trap you in gas rooms where you HAVE to shoot the door to escape as your guns won't - I think this voids the achievement, so it sucks if that happens). Just summon in two guns at a time to clear out each area and then wait for your ability points to recharge before moving on too far. This can make a level last a lot longer but it's the only way. Plus some of the bosses can be annoying to kill (especially if they are the type that summons in smaller goons). It also seems that if enemies get blown up by scenery explosions, or their own attacks, that this can void the achievement too. So it really is just RNG whether or not you'll get this first time.
Personally I did a random Data Purge mission with Nurgle Vanguard. As they have a lot of ranged enemies, and I could rush to each data point to hack it - then move onto the next. In the last room I just ignored the boss enemy entirely, snuck over to the last terminal - put a turret either side to distract the enemies that spawn - and then hacked it to finish the mission. The achievement popped as soon as that hack was complete.