If you want to tan that hide, keep the deer's brain. I know that sounds gross but it is necessary.
Get the hide off the deer, scrap the flesh die clean of all meat and tissue. If you want the hair off, soal the hide in water for a couple of days...the hair will start to 'slip'. Then scrap off the hair. Rinse the hide thoroughly. Mix the brain with warm water and make a semi-thick paste and begin rubbing it into the flesh side of the hide. If you are doing both side and have removed the hair, put the hide in a bucket with the brain solution and thoroughly saturate it.
The hide is going to smell funny and you may think it is going bad but it is not.
take the hide out of the brain solution, wring it out well and begin stretching the hide. This is known as softening. You are opening the fibers of the hide itself. You can resoak it in the brain solution at this point if you want a butter soft hide. Remove from the brains again, rinse lightly and wring and begin stretching the hide in every direction and as far as you can. DO not be surprised if your hands, shoulders and back start to hurt. Do this until the hide is completely dry.
Once the hide is soft and dry, Smoke it over a wood fire but be careful not to scorch it. Once the hide is smoked the color you want it, remove it and you have a braintanned hide that will last you 100 years.
For more detailed instructions, you can try braintan. com for info. I personally use a different process than he does but I do it the way I was taught as a child so, they do have good info on that site.
Here are some of my hides hanging around my deck in various stages of softening.
This is a small one completely softened before being smoked