Personally, if I were looking to do it for a 'throw' then you could do worse than taw it - basically using Alum (Aluminium Sulphate - about £10 for a kg on eBay) and salt. Most concise recipe I could find it this:
http://forum.downsizer.net/viewtopic.php?t=3498&start=0
It's a few years since I did any hair-on tawing but it works. Just salting it doesn't tan it, it just preserves before any tanning process by killing bacteria, so what you now have is hair on rawhide. You may be able to get a thick deer skin a bit soft just by stretching it, as the video implies, but the deer skin he was using was already tanned using bark. It needs some form of preservation. Tawing does this and the oils towards the end of the process allow it to get quite soft and pliable.
You could opt for a hair on brain tan, rubbing in brains or a substitute (eg. egg yolks) and then stretch and finally smoke it. The smell of the smoke does hang around for a long time, so if it is for your living room, I'd think twice.
Has the skin been framed or is it just been allowed to dry? If you are thinking of braining, I would suggest remoistening it and then put in a frame to redry. Having it flat make it easier to take out of the frame and then get the brains solution rubbed over.
Don't forget deer hair is brittle and will break so if you put it over a chair which his used regularly, the hair will begin to break and come off.