Not really pleased with the lack of information here so I'm going to post my own guide. The achievement is not guaranteed in the prison race for whatever reason. I have completed this game three times now, and 2 out of 3 times I won the race first try and did not get the achievement. As others have mentioned, you can try reloading your save and trying again, but to be honest I forgot to try that so here we are.
So if you are like me and missed it the first time, you can get this achievement much later in the game. What you will need to do is wait until you obtain the Highwind, so that you can fly over the the Chocobo Sage's house in the Northern continent. Look for a house completely surrounded with mountains and land there. The Chocobo Sage sells
Sylkus Greens for 5,000 gil each. I recommend buying
30-50 of them. As 150,000+ gil for all these greens is a bit pricey, try waiting until the very end of the game to do this if you are hurting for gil.
Now you need to head to the Chocobo Farm on the first continent (east side). Talk to gramps in the house and buy a stable for 10,000 gil. You only need one for this achievement, but you'll eventually need more if you want to get into breeding Chocobos for the Gold one.
As mentioned, you can skip this next section If you plan on obtaining a Gold Chocobo for the Knights of the Round achievement. But if not, head outside the farm to the tracks and equip your Chocobo Lure Materia. To catch a Chocobo, you need to kill all the enemies surrounding it without hurting the Chocobo itself. There's no need to feed it greens to occupy it as these low-level enemies will be dying in one hit in the end-game. After obtaining one, dismount and send it back to Chocobo Farm. You can catch up to four at a time to increase the odds of getting a "Good" or better one.
Once you've got a few, head back into the Farm, into the stables, and talk to Chocobo Billy. Ask him about moving Chocobos to the stable. Each Chocobo you caught will have a rating, and you want one that is
Wonderful, Great, or Good. These specific words will be in Billy's quote. If you got one of those, put it in your pen. Now talk to Billy and ask him to feed the Chocobo. You can now feed it the expensive greens you bought. I personally did 10 at a time but I think you can just stuff his/her poor face with all 30 at a time, lol. I fed it 30 before it no longer gained stats, but it could be dependent on the Chocobo. Speed is the most important stat, followed by Stamina.
Anyways, that should be all you need! Your Chocobo should now be a beast on the track. Head over to the Gold Saucer and register it to race with the girl on the left of the Chocobo Racing section. To actually win the race, you first need to hit
to switch it from automatic to manual, then hold
,
, and
. If you start to run out of stamina, hold down
for a bit but it shouldn't be necessary. The short track also works and is quicker. Achievement should unlock after just one race win here.
Before downvoting, please explain what I can add to this guide to make it better, or anything I may have forgot. I will be happy to make chances to help future players obtain this slightly annoying achievement. Thanks!