The achievement description is misleading.
You do NOT have to complete the entire level without being hit, that is virtually impossible, especially with all the cross bowmen.
At least for me this is what happened;
After retrying a few times after being hit, I decided to just move on & get a good idea of the level. When you get to the final boss I was even thrown off so again I did take damage (though if you're careful you may be able to not take damage), however when you defeat the boss, just like every enemy, you get a health orb. It replenished my health & the end cut scene initiated with both achievements for Combat Genius & completing mission 13.
Hope that clears things up for you, good luck!