To unlock Legendary Souls, you must first complete the Story mode on any difficulty.
Now I'm going to warn you, this definitely isn't the easiest achievement, so get ready to be frustrated as the A.I. is merciless and cheap in this mode. Fortunately, all you have to do is win one match by using a brave edge attack as a finisher.
What I did was pick Astaroth. My first match was against Kilik. As soon as the match started, I ran up to him and pressed A+X together to do a grab. Usually the grab would be successful. While he's down on the ground, run as close as you can up to him, and as soon as he gets up, grab him again! Rinse and repeat this method until you have defeated him twice. I managed to get a perfect on him a few times by doing this.
Now this is the tricky part. Once you've beaten him 2/3 times, you're going to want to repeat this same method until he barely has any health left. Once he does, attack him by moving Down, Down+Y. As soon as you hit this combo, hit RT. This will activate Astaroth's brave edge axe swing. As long as the attack is successful, you'll get the achievement.
NOTE: You CANNOT win just one round and expect the achievement to unlock (which I did). You must win the WHOLE ENTIRE fight, and finish the last round with a brave edge move.
Good luck, and don't break any controllers! Hopefully this method helps you like it helped me.