"Finishing moves" are combos you learn as the game progresses. You will have learned them all fairly early on. You can tell when you've taken out an enemy with a combo when you hear a disembodied voice shouting out a message (e.g. "Impressive!").
You can get this achievement if you finish off 5 enemies using any one of the three following combos:
Stun: A B B
Pop-Up: A A B
Trip: A B A
This one will pop pretty quickly if you just alternate A-B while wading through enemies.