>>2156906Peroxide and time like the other anon said. Bleach is really bad for bones like he said and it's everyone's first thought for what to use as far as I can tell. 3% peroxide from the store can work, but it can be slow and require more than one bottle if you don't dilute and/or have a large container. You do not need to whiten a skull to preserve it; I have plenty of whitened skulls and plenty of natural skulls. I have some that even have a little dirt still on them. Peroxide can be nice to have around; it will fizz when cleaning and will stop fizzing when it's clean or the peroxide is spent.
Degreasing is important whether you want a white skull or not. If a skull is greasy and fatty looking, you should soak it in some dawn dish soap and water. Never a good thing to have natural greases and oils going rancid in bone. Not all skulls require degreasing. Some have been sitting long enough that the oils are mostly gone, some were never that greasy in the first place, and macerated skulls may have had their oils stripped in the process of sitting in a bucket of water for weeks.
You can get stupidly white bones if you do it right and they can shine, too.