The head is fully skinned with a special technique called caping. Depending on the species, it can be a very complicated process especially around the mouth, nose, eyes, and ears. A taxidermist can do it for you, or you can check your library for do-it-yourself taxidermy books.
Most of those heads with the eyes, teeth, and so on are actually plastic and putty with the skins re-mounted carefully.
Turkeys are a famous exception. None of the fake heads look all that good, so tuekey farmers sell the heads off mature gobblers and they are freeze-dried, airbrushed, and sold ready made to taxidermists.