There are two pats to this:
1) Obtain the Cosmo Memory item from Shinra Manor in Nibelheim.
2) Obtain all of Red XIII's other Limit Breaks through combat.
I will be going over both of these!
Cosmo MemoryYou can on this on your second trip to
Nibelheim. You will be able to freely roam the town (or leave if you wanted to). Go to the Shinra Manor (the mansion) in the top left. When you enter, you'll notice a book to the bottom left. This will give you hints to a 4 digit safe combination. If you just want the code, it's below, but I'll also help out with the hints if you'd like to know how you come up with the safe combination.
Hint 1: The lid of the box with the most oxygen.
Hint 2: Behind the Ivory's short of tea and ray.
Hint 3: The creek in the floor near the chair on the second floor... then to the left five steps, up nine steps, left two steps, and up six steps.
*** Spoiler - click to reveal ***Start from the chair in the room with the elevator on the scond floor (try interacting or walking around until Cloud looks down at the floor). Follow the directions, counting Cloud's steps (use the D Pad to go slower/more precisely). You should end up in the hallway, finding the clue that gives you Right 59.
Hint 4:
So our final code is Right 36, Left 10, Right 59, Right 97.
Take this solution to the safe on the second floor, top left room. Use the D Pad to enter the code within 20 seconds. It starts from 0, and you can't go past the number you're entering next. If you mess up, just spam A, it will fail the lock, and you can try again.
When you enter the code correctly, you will fight
. This boss is Level 35, but in my fight, Cloud was Level 27, and Aeris/Barret were slightly lower. As usual, you can cheese the boss by using the "Battle Boost" (press Right Stick) and have infinite healing/Limit Breaks. If you want to do the fight legit, try Seal Evil (Aeris's Limit Break) or Cross-Slash (Cloud's Limit Break), as these freeze the boss for a short bit. Halfway through the fight, the boss will change. If it's Red now, it gains increased magic defense, and if it's Purple, it gains increased physical defense, so just use physical attacks if it's Red or magic if it's Purple. Defeat the boss and you will obtain Cosmo Memory! You will also get a key needed to recuit Vincent, who is also in this building!
Red XIII's Limit BreaksLike most party members, Red XIII gains Limit Breaks through combat. You'll need to defeat a total amount of enemies
with Red XIII getting the final blow, and use the first Limit Break in each level a certain number of times. I waited until I had The Highwind to do this, as it made things a little easier.
Here is a general sense of what you need to do to get Red XIII's other Limit Breaks:
Level 1
1a) Sled Fang- You start with this.
1b) Lunatic High- Use Sled Fang 8 times.
Level 2
2a) Blood Fang- Defeat 72 enemies with Red XIII.
2b) Stardust Ray- Use Blood Fang 7 times.
Level 3
3a) Howling Moon- Defeat 144 enemies with Red XIII.
3b) Earth Rave- Use Howling Moon 6 times.
For enemy defeats, I went to the forests near the Gongaga area. Sometimes you can find a group of six frogs, which will knock out a decent chunk every time. What I did is I would pair up Red XIII with one of your other characters that needs enemy defeats, and just use basic attacks and mash A throughout the battle. If a character doesn't need defeats, just have them defend. Turn on x3 speed by pressing Left Stick and you should be done within 30 minutes or so, depending on what you're finding.
For using Limit Breaks, I went to Cactus Island, just South and West of Cosmo Canyon. The island itself will not appear on the world map, but it's easy enough to find. You will need The Highwind to get here. I had intended on doing this somewhat legit, but the temptation of instant Limit Breaks from Battle Boost (pressing Right Stick) was too much. Before you find a battle, make sure you're on the correct Limit Level in the Limits menu. Find a Cactuer in the desert area, and just spam Limit Breaks with whichever characters need to use them. I was around Level 40 when I did this, and I found that even Limit Breaks were missing a lot of the time, which means more uses per battle! With this strategy you should easily get all of your uses done in 10 or 15 minutes.
Once you've gotten all of the Limit Breaks, and you have the Cosmo Memory item, go into your item bag and use Cosmo Memory on Red XIII. This will give him his last limit break, and your achievement!