Chocobo racing is based on the stats of your Chocobo. In order to progress up the ranks you need to feed your Chocobo Greens to increase their stats. To the right is a list of the effects of the different greens.
You can feed a Chocobo a number of Greens at a time, but be careful as they will stop having an effect after a while and you may end up wasting a lot if you are not careful.
After your Chocobo can not be raised any more, head to Gold Saucer and go to the Chocobo Square.