We employ lots of methods to do this job at university including the use of very dangerous chemicals. The methods work but the process is very hazardous and you would not be able to get the chemicals anyway. So what's left?
The first thing I'd suggest is finding someone with an ultrasononic cleaning tank. Immersing the skull at around 37 degrees C in distilled water in the tank for a few days will make quite a difference. A friendly, local jeweller might have the equipment you need.
The final method you can try is messy (which is why I've left it until last), but i guarantee it'll work. First of all. clean the bone by hand to the best of your ability, removing all detritus and soil etc. A toothbrush would be great but don't use anything other than water as a solvent. Place the cleaned skull in a bucket and pour tepid water into the bucket. Make sure the water level is at least six inches above the skull. Leave the bucket, covered with a piece of hardboard or something similar, in a warm place such as an airing cupboard or greenhouse etc.
Every few days, empty out the water from the bucket and replace it with fresh, tepid water. Be warned - hold your breath when you lift the hardboard as the smell is pretty unbearable as the liquid contains complex lipids and bacterial by-products that stink to high heaven. Carry on like this until the water that comes out of the bucket is clear.
Finally, remove the skull from the bucket, get some 10 or 20 volume hydrogen peroxide solution from your chemist and pour enough onto the skull in a small container, again to cover the skull. Be careful here as you need to watch what's happening carefully: the skull will gradually whiten before your eyes. When it reaches the right colour for you, remove it from the solution, soak the skull for an hour in water and rinse it well under a tap. The skull is the ready for use.