These can easily be boosted for in player matches, but here's how it works.
In order to get a Critical Finish, you must perform a Soul Break. To get a Soul Break, you have to beat the hell out of your opponent. At the end of the Health Bar, you'll see a circle that is usually green at the beginning of the round (can be blue because of certain kills). That is the Soul Gauge. When you attack someone, their Soul Gauge will deplete. If they block a lot while being attacked, their Soul Gauge will deplete faster. (Depletion is shown by the orb turning to a red color).
When the enemy's healt bar is flashing red, they are close to having their Soul Gauge broken. Keep on the offensive, but start paying attention. A glass shattering noise can be heard and their red Soul Gauge will turn black/gray. That is a Soul Break. The enemy will also have some small, purple lightning bolts flash around them to show what happened.
During this moment, you have to press A+B+G+K (X+Y+A+B) together at the same time (or LB with the default controls). If timed correctly, you'll perform a critical finish.
If you really just want the achievement and don't care about what character to use, I recommend Yun-Seong. He has a move that'll help you get this quickly. When fighting, press Y and then Forward+Y. He should do a vertical swing and then another vertical swing with lightning bolts (guard break). Keep doing this against computer characters (which will block a lot) and they should start flashing red in no time. Then, just hit Y once a second and a Soul Break should happen. Then press LB.