This is not actually possible. Any tattoo would be invisible under the fur in the same way that you can't see scalp tattoos unless someone's head is shaved.
Paint should blur and flake off as they move unless you applied enough of it that it just forms a solid block, and even then it would crack at the joints. (Though it would probably work better the longer the hair, since that would buffer the bending of the skin more. But if you do apply a lot of sufficiently sticky, flexible paint to fur, you'd better hope you have an excellent solvent to get it out again afterward.)
Permanent-ish color designs might work with short hair and dye, and even then it's not going to get you that good a level of detail because the longer the fur the more it blurs together.
I'm talking especially to you, author whose book I'm not really sure why I'm reading, whose werewolf main character apparently is covered snout to tail with impressive tribal tattoos in wolf form. Since your other main character is marvelling at them, I can only assume that the wolf also has no fur. Though come to think of it, you haven't actually mentioned anything about their fur yet, so maybe they really are bald.
Not-quite-related link! Why Paint Cats, a book about what it says on the cover, sort of.