There are several events that go into play in order to make this achievement possible.
1. In chapter 1 at Mushroom Rock Road, be sure to let Klasko onto your airship. You will need to go back to the entrance after completing the story there to talk with him.
2. In Chapter 2, at Mi'Hen highroad, talk to the girl and do the chocobo catching minigame. If done correctly, you'll get a cutscene telling you the girl is in trouble. Klasko should buy you some time while you run down to battle the chocobo eater.
3. Once you beat the chocobo eater, go back to the Mi'Hen travel agency and invite Klasko & the girl onto the airship.
4. Go to the calm lands (it should be a Hotspot now). Once you land, follow Klasko. This will take you to the Monster Arena from FFX.
5. You will have to kill fiends in the arena. There are several "fake" fiends and 1 real fiend at all times. You can find the real fiend by observing where the "fake" fiends are facing, as they will always point towards the real one. I recommend saving after each battle, just to be safe. Once you have beaten all of the fiends, go back into the arena and talk to Klasko to be given the greens needed to try and catch a chocobo.
6. I found chocobos to appear most frequently in the Thunder Plains & Macalania Woods. To catch one, throw a Gyshall Green at it as soon as possible, then kill the other fiends. Once it's alone, you should be able to throw additional Gyshall Greens at it until it is captured and the achievement pops.
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