No need to do any drilling, the spinal cord hole in the back of the skull leads directly into the brain cavity. You can use a stick, wire, or maybe just high pressure water to flush out that cavity.

Your skull will have to be completely clean before you put it into peroxide. The peroxide that you buy in the grocery store is only a 3% solution and isn't much good. My taxidermist told me to go to a beauty supply store and buy their 40% peroxide.

My taxidermist sends skulls to a guy that has beetles that completely clean skulls. He also boils skull caps with antlers, but that stinks.

When I hunted in Mozambique, the camp staff simply buried the skulls, and the bugs in the ground cleaned the skulls.

I once tried to clean a black bear skull by putting it on a large anthill under a bucket with a large rock on top, but the neighbor's dog found it and ate it.

There is a thread on one of the other hunting forums about skull maceration where you just keep a skull in warm (80*) water and the bacteria clean it. This method also stinks so you need a tight lid on the container, and it may take up to a year for it to completely clean.

You also might ask this question in the Taxidermy forum.