A "finishing move" is when you're fighting someone and a (fairly large) button appears on your screen (above the person you are attacking). Press that corresponding button and you will perform a "finishing move" on your opposition. It will appear after you have punched/kicked you opponent several times and they are falling backwards. When they are trying to regain their balance, that's usually when it will come up. You know you pressed it in time if your character begins a sequence of attacks and full-on destroys your opponent in a dramatic/cinematic fashion. :D So, 249 more times and you'll have this. Don't worry though, you shouldn't feel rushed, because this, along with several other achievements in this game will eventually come during your multiple playthroughs of this game.
Note: You also get more opportunities to perform "finishing moves" on the larger bad guys later on (ex: the huge bikers in Chapter 2).