While fighting your enemies you will be able to finish them off with a fatal blow once you've removed a limb using normal attacks. Simply press
next to an enemy once you've removed a limb and you'll see an animation of Ryu finishing them off.
Doing this 1000 times may seem difficult but it will come with natural progress. You will likely not earn this on your first playthrough, but try to pull off the obliteration kill whenever it's a possibility and you'll probably unlock this achievement sometime in your second playthrough of the game.