How to earn the “…Without Bullets!” 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.

An unofficial, but popular Deus Ex wiki says the following:

"The aim is to complete the game without killing anyone directly, or someone dying as a result of an action taken by the player. The only exception to this is the Boss fights...killing anyone in the Prologue chapter will lock out this achievement

And currently, it appears there is no way to see if you are still eligible for the achievement at any time:

"Unfortunately if you lock out the achievement by accident, unless you aware that you did so and reload an old save, there is no indication that you have done so and may end up playing through the game and not get the achievement due killing a character earlier in the game."

This seems to be a fairly sought after feature though. Since the game has been so popular since its release (over two million units sold already), I would not be surprised if a patch is released soon resolving these complaints.