"wash the dog"..is the normal way to talk about bathing a dog. If the dog has a specific part of him which is dirty you could talk about cleaning him up. But normally cleaning is applied to things like windows or clothing not animals.

Hi Victoria, We always say wash the dog. Clean his teeth and his ears:-) We also say wash the car, for the outside. Clean the car, for the inside. At least that's my experience. Hope this helps:-)