First, you need to go to the Chocobo Sage's house on the Northern Continent and buy as many Sylkis greens from him as you can. This will cost you alot of money. Then you need to capture a "Good" male and female chocobo to commence breeding. Feed them both 90 sylkis greens and race them to "S" class at the Gold Saucer. Save the game before you attempt breeding and feed the couple a Carob nut.
If all goes well, you should end up with a male or female "Green" or a "Blue" chocobo. Feed your new chocobo 90 sylkis greens and race it to "S" rank. Go back to the ranch and save your game. Now, use the breed another chocobo with the old couple that you used before. You should end up with an opposite colour and opposite gender chocobo than the one you got last time.
Feed the new chocobo 90 sylkis greens and race it to "S" class. Remember to save your game at this point. Breed your blue and green chocobos with a Carob nut, and you should end up with a "Black" chocobo. Feed it with 90 sylkis greens and race it to "S" class.
Get rid of the original parents and catch a "Wonderful" male chocobo. Feed it full with sylkis greens and race it to "S" class as usual. After this is done, breed it with your black chocobo with a Zeio nut. Do not forget to save before doing this. If all goes right, you will end up with a "Gold" chocobo!