I guess this was obvious to most others, but not me.
When you are in the shooting range, the far right position is a "sniper nest" showing three targets down range. You will NOT get the achievement for merely shooting the three targets.
On the wall to your right there is a sign that says "Activieren" with an arrow pointing to a red light. You must shoot the red light first to start the target mechanism. After shooting the red light, targets will begin popping up down range.
It's not too difficult to hit all the targets and get the achievement to pop. They give you plenty of time. I even had one reload and still got it.
Someone else posted this video demonstrating this. I take no credit for the video.