I have tanned two bear hides that I made into rugs but softened them so that they would be pliable enough to hang over furniture if I choose.
I purchased tanning supplies from Van Dykes. There are different tanning methods and chemicals so read through the options and select what will work best for you.
First: Scrape all the flesh off and split the lips and turn the ears, Then salt the hide and rub it in. Roll the hide up hair side out and set it over a grate or in a position that the moisture that the salt draws out will run off and not sit on the hide. Let sit for a day or two and then flesh and salt again. When you are satisfied that you have all the meat, fat and membrane removed you can either salt it and store it or freeze it or begin tanning.
Follow the directions for degreasing and tanning for whatever method you chose.
For softening I recommend using a frame made of 2x4's. Use 2x4's that will be long enough to give the hide room to stretch, for a six foot bear, 8 foot 2x4 will give you 1 foot of room around the skin. Lay the hide out flat, skin side up (easier to see the holes that you will poke around the edges). Poke holes every 3-4 inches around the hide and run small diameter nylon cord through the hole and around the frame to the next hole. The more holes you get through with a single piece of cord the better for saving time later while stretching( I try to get one length of cord for each of the four sides of the hide). Since I was doing a rug I also put the head form in place at this point so that as the hide dried it would fit the form and I would not have to do any stretching on the head. Once all the cord is wrapped through the holes and around the frame stretch it tight length wise and then width wise. As the hide dries, use a blunt narrow object such as the end of a shovel handle and push hard into the skin side of the hide and scrape in every direction. (I use a 1x2 board with rounded edges) At first while the hide is still pretty wet it won't do much but as it dries you will begin to see the stretching and feel the hide getting softer, pay close attention to the edges as they will harden the fasted if not worked enough and be careful around bullet holes and other weak spots so you don't puncture the hide. If you do puncture the hide, the good news is it is a bear and once you sew it back together the thick fur will hide the damage. If you are careful though you won't break through, raw and tanned skins are very strong, I never have and I've done 2 bear 3 deer and two elk this way. After each stretching session tighten the cords. I usually stretch twice a day, morning and night. After each stretch I use a spray bottle to spray oil on to the hide.
Depending on how good you did when you fleshed the hide there may still be some membrane attached, when the skin is dry enough you can sand it off by using the foam sanding pads that are made for drywall. THis will help bring it to a soft suede-like texture.
