A coup de grace is a special physical attack which can be performed on enemies under certain conditions. Usually it involves dazing, or staggering the enemy. The easiest ways I found of setting up this attack are:
On regular enemies (zombies/J'avo) - Flash grenades will daze enemies in a way that will allow you to follow up with a coup de grace. Throw a flash grenade into a group of enemies, then run up next to them and hit when prompted for a coup de grace. If you're quick, you can get 2-3 in a group before they recover.
On B.O.W.s - On tougher enemies (most commonly creatures that come from cocoons) they'll fall to their knees once they've taken enough damage. Run next to them and press for a coup de grace.
If your partner is grappled - If your partner gets grabbed, rescue them by pressing the and you will usually perform a coup de grace.
You'll most likely unlock this just playing through the campaign. If you somehow need a few more, an easy place to score some coup de graces is at the beginning of Chapter 4 in Leon's campaign where you fight a Leposita. Damage it until it kneels, and up close and repeat until you've unlocked the achievement.