You must execute 10 criminals.
Criminals are the enemies that have their own life bar. One will always appear at the end of the Last Encounter (Wave 5), the only other time they will appear is during a random challenge while playing. In the ladder case, you must kill/ arrest him before killing all the other enemies in the wave or he will simply disappear.
To execute a criminal you must first deplete their life bar, at this point they will hunch over and you must either kill or arrest them. To execute them you must deplete their life bar completely. Their life bar will gradually regenerate if you are to far away from them, so I suggest getting right next to them and unload on them so their health doesn't regenerate faster than you can deal damage.
Killing the criminals in Commonwealth Tower does not count towards this achievement.