There is a lot of RNG in chocobo breeding.

The first priority, get your stars in order.
To increase the stars - purchase P9 breeding. These have the highest stars available. You will need to buy a bunch of them and be ready to throw several away. Basically, you are using GIL to offset the RNG of the star disbursement.
You cannot buy a 20/20 breeding I think it maxes out at 14 or there abouts. But as 14 you will have usually 2 or 3 4* stats and a couple 2* star stats.

Once you get a decent bird star wise all stats 3 or 4 stars, you need to start breed that bird with its offspring. This will lessen the star RNG by having two parent birds with good pedigree stars.
During a couple of mid pedigree cycles you might need to return to the GS P9 breeding to fillin the gaps your 4* pedigree.

After the stars are taken care of you just need 1 or 2 more breeding cycles to set up your inherited skill and fair/foul weather traits.

My birds for example, have had terrible RNG, so I have been stuck at about a 17/20 star bird for many breeding cycles. I am considering starting over.