This trophy has a few prerequisites to be able to get it.
First, in chapter 1 when you go to Mushroom Rock Road, after doing the events there go back to the entrance and talk with Clasko and let him ride your airship. Be sure to talk with him after letting him on board.
Then in chapter 2 go to Mi'ihen Highroad and go inside the Travel Agency on the northern end (you can walk or take a hover, walking is good to get gil and AP). Then approach the girl who asked about chocobos outside on the cliff. This will start a rather annoying mini-game to catch a chocobo. While this catch doesn't net you the trophy, it is required. Then once you finish the quest a man will come running and say the girl is in danger, go off and rescue her. Clasko will come and buy you some time to run below to rescue her then you have to fight the Chocobo Eater from FFX, though this battle is much easier.
Then after that is done, go back to the Highroad and talk to Clasko and Calli, invite both of them and the chocobo onto your airship. Then the Calm Lands becomes a hotspot. Head there and then you get to do some fiend hunting.
The fiend hunting mission sucks solely because of the Queen Coerls. Have lots of phoenix downs and hi-potions handy, because you'll use them. Once you beat the fiends this opens up the ability to catch chocobos for Clasko. Go back in and talk to him and he'll hand you a bunch of Gyshal Greens and explain how to catch them.
Then it's just a matter of finding one in a random battle, feeding it greens, defeating the rest of the fiends and then feeding it greens again to catch it. You'll have the rest of chapter 2 and all of three to pull it off which is plenty of time. I'd recommend Macalania Woods for trying to find them, I seem to get the most battles with a chocobo in them in there.