You will not get the achievement if the enemy sentry is in the construction process and it must be a regular sentry. Your sentry on the other hand can be any kind.
The Wrangler allows your Sentry Gun to
shoot at an enemy Sentry Gun from
outside of its range, unless the enemy
Engineer is using the Wrangler as
well. On the offense, try to haul your
Sentry Gun into a position where you
can easily attack the enemy
Engineer's, the element of surprise is
often enough to force other players to
abandon their building nests. Unless
your Sentry Gun is level 3, try to
make sure the enemy Engineer is not
nearby, or kill him or her before
attempting to destroy the Sentry Gun,
as the Engineer will be able to repair
any damage done to it otherwise. Be on
the lookout for other Engineers trying
to deploy their Sentry Guns around
corners and use the rockets of a level
3 to destroy them. Destroying a Combat
Mini-Sentry does not count towards
this achievement.
http://wiki.teamfortress.com/wiki/Obtaining_Engineer_achievements
Right, I've hit the achievement, and here's the info:
I've now counted 16 monitors in Chapter 8, but you can ignore the first 5, so you need to break all the monitors from the one found in the test chamber numbered 1/19
, up to the start of Chapter 9.
Instruction for breaking each monitor:
Chamber 1
Move the cube to the lower section of the level. Use a ceiling-floor loop to get the cube to high speed then fling it from the opposite wall to the monitor.
Chamber 2
Move a cube to the ceiling using the vertical funnel position that forms the normal solve of the level and drop the funnel to fling the cube at the monitor from the wall in front of it.
Chamber 3
Another cube fling, either using a ceiling-floor loop or funnel to get the cube up to speed for the fling.
Chamber 4
You need to sneak round to behind the turret (I managed to run to the wall next to it before it could kill me, wait and then run again) and carry the turret towards the monitor. As long as you do this with the turret being "OK" (ie, don't let it shut-down) then it will target and shoot the monitor for you. You might have to put it down (as long as it lands correctly).
Chamber 5 (has 2 monitors)
The closest monitor is broken by flinging a cube from the opposite wall using a vertical funnel in the centre. Place the launching portal directly under where the funnel hits the wall to ensure you're on target.
The second monitor only appears when you reach the exit door, and is broken by the same method but by flinging the cube from the far-right space on the angled launch pad opposite.
Chamber 6
Just step on the aerial faith plate to launch yourself at the monitor.
Chamber 11
Once you've set up your laser (for completing the level) instead place the exit portal on the wall facing the monitor near the target point for the laser. Then maybe tweak the angle of the laser with the box to point it at the monitor for a few seconds.
Chamber 12
When you're setting up the laser to aim at it's target, just point it right a bit (at the monitor) for a few seconds.
Chamber 15
I did this with a double-fling of myself. Set a on the opposite wall and one of the floor below it, step in to the one on the floor then fall in to the one on the floor again - this second fall should give you the speed to fling across for the break.
Chamber 16
This one was quite difficult, but basically you have to get hold of a turret and get that to shoot the monitor (as per chamber 4).
First, I used blue gel to remove two of the turrets (instead of all three as you would to solve the chamber). Then I carefully placed a portal on only half of where the funnel is hitting the wall and linked it to one on the floor - this means the funnel doesn't "pass through" the portal, so you can effectively sit on that wall. From here you can shoot the other portal in to the wall above the turrets, then reposition the portal that's misaligned on the funnel's wall to let the funnel through. Then you can ride the funnel and drop down on top of the turret.
Additional monitor
Once you've started chapter 9, when Wheatley tells you that you're getting close to his lair, you can blow up the monitor using the falling bombs (this is the room that he tries to kill you with the spike plate and conveyor belt).
Please note that this does not contribute to the achievement.
Best Answer
Reading up on the achievement many people suggest getting a considerable distance away from the golden slime and hitting it with food so it is less likely to notice you being there. They also suggest try being a higher elevation than the slime.
If that way doesn't work there is another way that I read about on the steam forums for the game.
Taken from: https://steamcommunity.com/app/433340/discussions/0/358417008725609519/
So if the first way doesn't work the second option may be your best bet.
Good luck those gold slimes are very crafty.