There was an eHow article (now archived) detailing how to unlock the achievement, but the basic steps are as follows:
In stage 2 of the Dark Carnival, score 750 points on the shooting game before time runs out. Upon winning, you get Gnome Chompski.
Your party then has to carry Gnome Chompski into the escape vehicle at the end of the campaign. You can drop the Gnome as much as you want, and any player can pick the Gnome up, but you must carry him to the end.
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
You just have to play 42 games. In the leaderboards there's a section called "Games played" that tracks it.