You will find the Carbine first time during the Soldier's Field level, inside the locked room to the left of where you start. It requires one lockpick to open, and the weapon itself rests near a table in the very back.
Enemies will also carry the weapon from this point on.
Remember, kills using the Carbine and the Burstgun (available some time after the Vox are introduced) both count toward the achievement.
The kill count - as with the other weapon achievements - carry over to different playthroughs.