This achievement is for defeating every character in a run of arcade mode using Elisabeth. This means that all 5 sets of 3 characters are defeated using only her, you may keep a second controller on hand in case you come close to losing a round in order to save yourself.
Elisabeth is one of the tougher characters to do this with. She has no fancy projectiles or anything of the sort to carry her through the arcade mode. It's because of this the only real method I could come up with was playing on simple mode and trying to jump in with a heavy punch (Y) and then linking that with a down + forward + Y special where she swipes her hand like a blade. It's simple, but it was the only decent means of damaging I could find. Also, this is nice because her ultra is a concentrated blast of energy that takes up approximately 2 character spaces in front of her. It's a large ball that stands still, so you want to be close when using it. You can execute this with down , forward + punch when the meter at the bottom of your screen is full.
As with every character, it is recommended you keep track of who you defeat as you must defeat the whole roster of characters in arcade mode for your King of Fighters achievement.