In order to get Red XIII's final limit break, you need to unlock the six prior limit breaks and obtain the item that teaches Red XIII his final limit break. The way to unlock each limit break differs for each character but are essentially similar. You unlock the second limit of each stage by using the first limit of that stage a bunch of times, and you unlock the next stage by getting a certain number of kills.
Red XIIIs limit breaks
Stage 1: Sled Fang, Lunatic High
Stage 2: Blood Fang, Stardust Ray
Stage 3: Howling Moon, Earth Rave
Stage 4: Cosmo Memory
Unlock order:
Use Sled Fang 8 times in order to unlock Lunatic High.
Get 72 kills with Red XIII in order to unlock Blood Fang.
Use Blood Fang 7 times in order to unlock Stardust Ray.
Get 72 kills with Red XIII after unlocking Blood Fang in order to unlock Howling Moon.
Use Howling Moon 6 times in order to unlock Earth Rave.
Use the Cosmo Memory item after unlocking the six previous limit break in order to unlock Cosmo Memory.
Getting the Cosmo Memory item is relatively easy and straightforward. You just need to unlock the safe in the Shinra Mansion in Nibelheim where you fight the Lost Number boss (the same one containing the Odin materia, and the key to the room in the basement where you find Vincent.) The combination is right 36, left 10, right 59, right 97. If you go over then you have to start again, and you only have 20 seconds to enter the combination correctly.