Vincent's Limit Breaks:
Level 1 - Galian Beast - Starting Limit Break
Level 2 - Death Gigas - Defeat 40 enemies in battle
Level 3 - Hellmasker - Deafeat 56 enemies in battle after obtaining Death Gigas Limit Break
Level 4 - Chaos - Use the item Chaos
To obtain the item Chaos, Vincent's final Limit Break, an Ultimate Weapon and this achievement you must do a small side quest which can done after completing the Underwater Submarine Battle and obtained the Shinra Sub vehicle.
Take the sub and travel above water over to the peninsula where Costa del Sol is located. Go in behind into the waters on the west side of Costa del Sol and look for a river with one grassy side on the left and a rocky side on the right
Submerge the Shinra Sub and you should find a cave nearby.
Bring the sub back up to the surface when you reach the waters on the other side of the cave and then dock your sub on the rocky ground on the west side of the pool.
Add Vincent into your current party and then enter the cave for a cutscene.
The game will automatically place your party back outside of the Cave once the dialogue has concluded.
After engaging in 10 battles return to the same Cave with Vincent in your party again to retrieve his Ultimate Weapon, Death Penalty and his Level 4 Limit Break, Chaos.
I've also attached a video below just in case you need help locating the cave mentioned above.
Credit to Jegged.com for images used.
Credit to Outer Heaven Guides for the YouTube video: