Calcium Carbonate is an absolute necessity and can be purchased by the pound at a pottery supply house... bleach should never be used with skulls.
Never used Calcium Carbonate (which is just Chalk?) and I thought it was only used to form a paste with the peroxide, making it easier to apply?
I generally simmer the heads gently until the meat goes soft. I've tried adding Sal(washing) soda (sodium carbonate) and/or dish soap and in future I think I'll go with just dish soap.
The trick is to change the water two or three of times during the simmering process to get rid of the accumalated grease and other crap.
As you change the water and also right at the end, as you lift the skull out, rinse it off with clean boilling water from a kettle to get rid any surface grease/gunk...
For really greasy skulls like hogs or bear, you can then soak the skull in white gas for a week or so to leech out as much of the remaining grease as possible...
Once the skull is clean and free from all meat, but before I soak it in peroxide, I dip it back into boiling water for a couple of minutes. I then soak white cotton wool balls in the peroxide and completely cover the skull. The skull needs to be in a plastic container to catch the "run off" and this also allows the cotton wool to keep wicking up the peroxide...
Not tried heating the peroxide or adding peroxide to the boiling water as the above method works fine and gives porcelain white results in about 24 to 36 hours. In fact, even after the skull has be rinsed of all peroxide it will continue to whiten if left in the sun for a day or two...
If done this way, its fairly easy to get results like this Muntjac skull below:
On the other hand, if too aggressive chemicals are used for degreasing, or ordinary bleach is used instead of peroxide for whitening, you risk getting pitting in the bone as below:
One other tip is at the start of the process, once you have the head removed from the carcass, roughly skin it out and leave it sit in a bucket of cold salted water for at least 24hours or until you get around to cleaning it properly.
The cold salted water leeches out any blood and effectively is a first step in the cleaning process. I've tried both ways and it does seem to help a bit....