You must complete Desperate Escape with an "S" rank on any difficulty. There are several categories that you'll be graded on, depending on your performance. If you average out an "S" rank, then the achievement will unlock.
Getting this with "Way of the Warrior" works well, because most of the time if you get 150 kills, then you an get "S" rankfor enemies killed. If you can keep you Accuracy high enough, then you should receive an "S" rank for that, as well. In order to get an "S" rank, at the end, your total score has to equal 80,000 or greater. So if you do not too good in one area, but REALLY good in another, it still works.