If your skull is old and already stained with grease from leaving in contact with flesh and skin for too long, the discolorations will never come out.
If your skull is fresh, here is what I use for small mammals with museum quality results:
Remove skin and as much meat as possible (don't cut skull with your razor).
Next remove as much of the brain as you can through the spinal column hole at the back of the head.
Place the skull in a pot that will allow you to COVER all of the skull with 3" over the top (important to keep a ring from forming where the bone sticks out of the water.
Cover skull with tap water and a cup of Dirtex or Soilax. This is commonly available from your hardware store or maybe even the grocery. The soap will turn the water greenish yellow but will NOT stain your bone.
Bring to a boil, then simmer (outside for gods sake!).
When you get a dirty looking foam cap and the greenish water looks more grey than green, change the water with fresh tapwater and another cup of soap. If you keep your fresh water real hot it will save you some time. What you are doing is digesting the cartilage and wicking the oils out of the skull.
I simmer for 2 hours with 3 water changes. The teeth will fall out. Not a problem. You will want to glue them back in with SuperGlue later.
After 2 hours drain the skull and use a soft tooth brush and a plastic pick to remove the now jellied meat and cartilage. Do this while still warm.
Now your skull is ready to be covered in standard peroxide for 24 hours. Standard peroxide is readily available, cheap, will not damage the bones. It takes longer, but it is safer for the project.
Keep the bones covered or you will get "rings" of high bleach and low bleach areas.
When bones are white, white, white, shake, drain and let dry. After 24 hours, glue teeth back in. I use a thin coat of spray-on shellac to make dusting easier, and to avoid stains from being handled.
In the picture you can see one old skull in the middle of a bunch of new skulls. The new skulls I did from animals I killed. The old skull was found in a field. I wanted to see if I could bleach it out...I couldn't.
Kris in NY