The achievement will only unlock once you have achieved 100% for the zone.
The achievement description would make you think simply completing the levels would be enough, but each level has 4 star objectives, those are what needs to be completed in order to unlock this one.
The good news is, (though this is only tested in the first area), that the goals are accumulative, so you don't need to meet the goal in one run, you can rack up the 20 headshots over multiple runs, so no stress
Hope this helps