The Orange Box

Once you have left the mines with Alex and the vortiguant, you will be attacked by regular and acid antlions around a bunch of repellant towers. After a bit of fighting and moving through the area, Two antlion guards (The biggest antlions that only crop up now and again). Just kill them. Now this might be a bit harder because you don't have the pulse rifle yet. Your best bet is the shotgun. Just keep pumping them, focus on one then the other. The achievement unlocks after you kill them both.

Twofer: Defeat both antlion guards outside the White Forest (secret).

This is story related in Half-life 2 Episode 2. At the start of chapter 3 'Freeman Pontifex' you pass three antlion seismic machines to get to an elevator. However, before you can get to it you have to kill the two antlion guardians that spawn out of the mine shafts on either side of the valley.

There's plentiful explosive barrels lying around and I found these the most useful weapons. The final seismic machine is surrounded by metal girders and boulders. The gaps were big enough for Freeman to get through but too small for the antlion guards. My strategy was to prepare in advance and I piled all of the explosive barrels by the seismic machine before triggering the antlion guards. I then stayed by the machine whilst firing barrels at the two guards. I found that I took minimal damage this way and didn't use any ammo either.

You can use your normal weapons to kill them as well. This video by luciferosx shows his strategy using normal weapons and barrels. He used the same area to hide in as I did.