You can buy or breed one, if buying check the like statistics thing under it if that makes sense

It should say the breed and then purebred, if it doesn't say purebred it is not

To get a purebred foal through breeding, you have to cover a mare of purebreed with a stallion of the same breed that is also purebred. You can find out if your mare and stallion are purebred by checking the Genetic tab on the horse's page and seeing what breeds are listed. To be purebred, a horse must list 100% for one breed.

If you use a mare but cover her with a public covering offer, you need to make sure to check "purebreed" (the blue star) on the covering offer page.

Use the filters to select purebred when buying and breeding and check that the female you are breeding is a purebred

When looking for a purebred in sales, there's a blue star check box to click. This will show the purebred horses.