Here are some instructions. You will have to degrease it first by boiling in dish washing detergent, http://www.adfg.alaska.gov/index.cfm?adfg=hunting.skull

If it was a metal bucket and those are rust stains then there is likely nothing you csn do, but for degreasing i would recommend dish soap and absolutely NOT boiling it in washing detergent because too much heat may degrade the bone and affect the longevity and additives in the detergent may be abrasive.

More likely the brown color is from the tannin in the soil carried into the tub by the water.

Get 40% ( by volume ) peroxide from a beauty supplier like Sallies Beauty supply. Mix it with Whiting, that you can get from Van Dykes or McKenzie Taxidermy Supply. Make a paste and, using a paint brush, cover the entire skull with the paste, wrap it in saran wrap ( to keep the paste from drying out ) and put it in the sun or under a really bright light or a UV light such as a tanning bed. After a couple of days, clean it off and repeat if necessary.

I have no knowledge on the art of taxidermy but this guy has some good info on skull mounts :

For anyone watching that, this guy has some really horrible advice. Including deliberately damaging a skull because your method of cleaning is poor. To get those mineral stains out of your skull, you need to soak it in a solution of powdered Iron Out. 1/4 cup to 4 to 5 gallons of hot water. Let it sit for a few hours and when it looks good, remove it and rinse well. Don't ever heat or boil peroxide. Huge waste of product. Soak your skull in straight peroxide overnight and it will be as white as it can get as long as the grease is out of it. You might be able to force it a little more by pasting it as above with either Basic White (also from where you get the 40 VOL peroxide) or at the least, talcum powder. Wrap it with plastic wrap and get it under a heat lamp or into a very hot area. UV has nothing to do with pushing the peroxide to whiten more, it's heat. After baking it for 24 hours, rinse well in hot water and it won't be any whiter. If you do use the Iron Out, do not use peroxide after doing so. Your stains will return with a vengeance. I made a post about this with photos on a different taxidermy board.

If at all possible, macerate/rot your skulls clean. Boiling causes a lot of other issues and you will take more time trying to correct those than if you rotted it clean to begin with. Maceration of a deer skull takes 2 weeks at the most if done correctly. In addition, a lot of the oils and grease in the skull are broken down by doing it this way and finishing it is a lot easier. Boiling will cook the grease into a skull and you will pretty much never get it all out.