The key is to cool the meat down as fast as you can.
Field gut it and get it to camp and hang it up in a shady place. Wash out the inside with cold water, this helps to cool it down. Cut the legs off at the knees. The idea is to make the pieces small enough to get in a cooler with ice so it can cool down as fast as possible.
Skin it as soon as you can, some friends of mine will skin it with a tennis ball and a winch on their jeep, lol. Make a cut around the neck up by the head and slit to all the way down to the belly, loosen the skin enough to get a flap of skin to fold over a tennis ball and tie a rope around the skin below the tennis ball. attach rope to winch and as you winch in the rope the skin will come off. Use a knife to ease any areas where the skin is sticking or where the muscle is coming off with the skin. Be aware that there is a very thin layer of muscle right under the skin that the deer uses the "shiver" off biting bugs, that can stay with the skin.
First cuts are to remove the tenderloins and back straps. Those are the best cuts of meat in my opinion, so treat them well.
Cut the leg muscles where they attach to the body, working tight against the bones. Most joints can be broken once the surrounding muscles are cut. Leg quarters go into coolers with ice, coolers go into the garage or under the trees to be in the shade. Front legs should have most of the shoulder meat on them, and the hind legs have all the hams.
We go for boneless meat so we remove everything we can from the remaining carcass before working on the legs. Meat from the neck goes into a bowl for ground meat. You determine how picky you are going to be about cleaning the carcass off.
My sister and brother-in-law remove the ribs and BBQ them for supper, too much bone to bother throwing into the freezer. Takes up too much space.
Front leg meat can be pretty tough, we use that for ground venison.
Hind legs, piece it out into steaks and roasts which are the size you want. I am very picky about my venison and I remove all the fat and silver skin that I can.
I then take the bowls of meat into the kitchen to sort them out and cut them, vacuum seal, label and freeze.