Breeding animals in Zoo Tycoon is very easy, you'll need spare animal capacity in whichever exhibit you're using and male and female animals of the same species. Animals will only breed if they are happy, so that is a pre-requisite for colour variants too.
Colour variants are a special skin pigmented version of these animals, and there is a colour variant version of every animal you can put in main exhibits in the game. Though whether a baby bred in your zoo is a colour variant or not is chance, there are things you can do that greatly increase your chance of producing a colour variant.
The most important thing is ensuring you have the level three breeding research in your zoo, which increases your chance of breeding a colour variant by 50%. The other key thing is having a three-star breeder from a breeding centre assigned to each exhibit where you want to breed colour variants. I've also had higher success breeding colour variants when the zoo has an abundance of zoo keepers, so I'd recommend having a good number of three-star zookeeper too. In general, once you have a good exhibit with spare capacity set up and a breeder assigned it takes between 10 and 30 minutes to get notified of a pregnancy.