Jersey Fresh recommended a bucket heater to do this task, so I got one on Amazon for forty bucks. You farm guys might know this, but I didn't: a bucket heater is used to keep the water buckets and troughs from freezing over for livestock. You plug it in, put it in the bucket, and it heats up the water. Easy.
I removed the skin from skull--not so easy, a bit tedious, but not impossible; removed the eyes and brain--again, a bit harder than you might think, a tad grisly, but nothing a hunter with a sharp knife can't handle; and I put the head in a plastic 8 gallon water bucket (the kind available at Home Depot or anywhere), along with the bucket heater. Filled with about 9 inches of water.
Worked like a charm. The water never boils but it gets plenty hot; tie the skull so the antlers are out of the water; periodically refill as the water evaporates over time. Neither the water nor the bucket heater got anywhere too hot for a plastic bucket, it was fine, I thought I might need a metal bucket but I didn't. You're essentially doing a slow, soup-like cook of the meat on the skull--smell is kinda funky, do NOT do any of this indoors. Your wife will kill you.
Process took longer than most guys said it would, here and elsewhere in the Internet--it was late November and cold out, so maybe that contributed--but I had my deer skull in the water for at least 9 hours over two days. Still, I was in no rush. Even with the hot tub bath, you still have to do a fair amount of scraping. And be careful, those deer teeth are sharp!
Re removing the brain, lots of Internet nonsense out there in my view, but this worked for me: I cut the skull, at an angle, from behind, using a simple coping saw; removed a small portion of skull (maybe 2 inches by 3 inches) that you wouldn't see anyway from the front, but it opened up the brain cavity so it could be more easily scrambled and removed. I used an wire hanger then a hose, but be careful--you don't want to use too much water pressure because deer brains all over everything is not only messy, but potentially dangerous. Be aware of CWD and prions and what they can do to you; clean everything afterwards using a Clorox-water mix of 1 to 4 or, better yet, toss out the saw blade, they're cheap.
Didn't use peroxide to whiten--a good tip, and one I've used on turkey spurs; instead, I hung it on the garage wall, where it's been for the better part of a year. White enough, really cool looking. And cheaper than a taxidermist; a great technique for a deer that's just not worth hanging on a wall. Mine was a 4 pointer.
Oh, I looked into The Buck Boiler for a hundred bucks; it's a plastic bucket with built-in heating element and auto shut off. I was not impressed with some reviews; the auto shut off for some guys was too easily triggered. And the bucket heater was 40% of the price, so what the heck. I essentially did the same thing with a bucket and bucket heater.
Hope this helps; I know Jersey Fresh's comments last year were a big help to me.