Here's my suggestion, based on the actual circumstances that earned me the Specialist achievement.
Play defense on pl_badwater. Make sure your team has a few competent damage sponges (Heavies, Soldiers, and Demomen) who know how to retreat (you may need to explain that bullets are hazardous to your health).
You want to stay alive. You can facilitate this by always sticking to a buddy. The best buddies to stick to are the ones that hang around sharp corners near narrow corridors:
In these locations, you and your buddy can stay behind cover in the yellow area, and once they are fully healed (to 150%, conditions permitting), they can venture into the red zone for ideal damage exchange along the highlighted lines of sight (where Heavies, Soldiers, and Demomen have a captive audience, so to speak, of the cart's escorts).
Note that the general advice here (be on a good team, play against a bad team, heal everybody) still applies, and that you'll still need classes like Engineers (who can help by putting dispensers near you) to cover your back.
Best Answer
There's no in-game way to track your progress that I'm aware of.
That being said, you only need to kill five. It shouldn't be that hard to remember which ones you've killed, given a list to look at. Nukapedia has a list of possible targets on the page for this achievement.
Note that only Super Mutant Behemoths and Mirelurk Queens count, and that you need to land the killing blow yourself. Here's the list, which doesn't appear to include DLC:
Super Mutant Behemoths
Mirelurk Queens