First, make a cut from neck to nethers and pull it open. Scoop out the intestines and throw them as far away from you as you can. This will stop the gut swelling from contaminating your meat and give the local scavengers something to occupy them so they don’t bother trying to steal the deer from you.
Next, peel off the skin. There are some ligaments you’ll have to cut through, but for most of it, you can sort of punch at the underside and it will come off. Bonus: you now have a hide. If you’ve read a lot of prehistorical fiction, you can figure out that if you tie the corners of it to some sticks and put it in the smoke from your fire to dry it out. You can then use it as clothing or carrying sack or something to sleep on. If you’re feeling really ambitious, you stretch it as it dries and work the deer’s brains into it, which will make it softer and more bendable, increasing its usefulness.
All the skeletal muscle is good to eat. Roast some of it for an immediate meal, and cut the rest into strips. Hang 'em over your smoky fire and you have venison jerky, good for later snacking or reconstituting and making into broth.
I’d avoid eating the brains, as I don’t know if deer carry prion diseases or not. Besides, I’d want the brains for the hide, as mentioned above.
I wouldn’t know which internal organs are good to eat, so I’d probably leave them to the scavengers unless we were in danger of starving. If that was the case, I’d choose the kidneys first, then the lungs and the liver.
Stomachs and bladders are good for holding water, so I’d use them to make rough canteens. I might try to figure out how to use a vertebrae to make a stopper for them, but honestly I’ve only read it described, so I’m not sure if I could do it or not. But a strip of sinew can be used to tie the opening instead.
If your deer has antlers, I understand they can be used as shovels or digging tools.
I wouldn’t know what to do with the rest, so I’d drag it over to my discard pile for the scavengers.