Kafei and Darkie
So, you wanna perfect your Chao, eh? There's no real need to have a Chao with all S-ranks - you can beat everything in the game with C-ranks if the Chao is level 99. But those shiny gold S's just look so cool... and there's nothing like watching a Chao with 3100 Run tearing up the field in the Peridot.
Getting a Chao with all S grades takes the longest time out of anything you might do with Chao. Be warned.
To start with, evaluate the Chao you have as breeding stock. See if any of them already have an S grade in anything, or even an A. If you have a baby Chao with an A in something, evolve him to that stat and the A will be upgraded into an S. For instance:
baby Chao Soap has Swim A, Fly B, Run S, Power D, Stamina C.
When Soap is evolved to be a Swim Chao, the A is upgraded: Swim S, Fly B, Run S, Power D, Stamina C.
Another Chao, Mallow, has C, B, B, C, S after evolution.
Breeding Soap and Mallow together has the potential to result in a Chao with S, B, S, C, S!
Unfortunately it doesn't always work like that. There's an element of randomness in Chao breeding that can be infuriating.
To be able to try again and again without spending massive amounts of rings, buy two heart fruits. Ideally you have your two breeding Chao alone in a garden. Feed the fruit to the two Chao, then when they are both ready, leave the garden and save. Turn the Gamecube off at the power switch.
When you turn the Cube on again, go to the Garden and your Chao will both be in season. Put them together and hatch the egg. Take the baby to the doctor. If it doesn't have the stats you want, turn the Gamecube off at the power switch. Repeat.
Chao breeding gets especially weird when you've got two Chao with very good stats and are trying to breed from them.
Kafei and Darkie
Kafei, a Duotone Purple with grades S B S S S and Darkie, a Duotone Black with S S S S A, mated. The baby came out as C S S S A, a step down!
If you get a baby with four S stats and a B or C, it's best to keep him and upgrade his bad stat yourself. My S-rank Chao Chi was born with S, C, S, S, S, but after repeatedly evolving him to Fly, that C moved up to another shiny golden S.
There is, so people say, another way to get an all S rank Chao. I am currently doing it with my silver Chao Sirius. You simply keep reincarnating the Chao and upgrading a stat each lifetime.
From C, C, A, B, D, Sirius has improved to A, B, S, B, A in seven lives. Some people suggest standing your character in the water and just leaving the Gamecube on for hours to age the Chao. Note that leaving the Gamecube on for long periods can damage it!
I do not advise doing it this way. XD It is easier to breed.