How can you earn the Specialist achievement?

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.

You really just need to play extremely conservatively, preferably on a long map that favors static positions, ideally some standard (not attack/defend) cp_. Overhealing does count as healing, but the best way is to try to heal people that are retreating from the front as they heal faster than those that have recently took damage.

As far as weapons to use, the new bonesaw that heals on taunt may help a tiny bit, it heals almost as well as the Mediguns, though it doesn't overheal.

A final note, your ability to heal is highly dependent on your team. People who play on pubs where nobody goes medic will play differently, just running to their death, instead of trying to get heals when low. I have no actual evidence of this, but I would assume servers with instant respawn encourage such bad behavior. If you play on higher quality pub servers, you may have better luck.

Heal snipers on the battlements on ctf_2fort. This is how I got the achievement.