To avoid cutting too much skin, skin from the back legs to the head. Cut the front legs at the elbow and peel skin back far enough to pull the legs through. I would allow the carcass to cool before quartering. I can fit a 130lb dressed deer in a 72 quart cooler, covered with ice. I do all of my processing in my kitchen but all preliminary work is done in my basement. Have a clean rubbermaid tote full of cold water to put the finished cuts in. You should rinse the cuts with clean water before wrapping. Don't try to make your cuts like you see in the grocery store, it's a waste of time. Here's my steps:
Harvest animal. Field dress, removing all internal organs and split pelvic bone. Try to rinse the internal cavity if you can. Remove the tenderloins ( Best part ) These are located inside the cavity toward pelvic bone. Skin the animal. If buck (mountable) from back legs to the base of the neck. Remove as much fat as you can!!! Easy way to remove hair stuck to the meat is with a propane torch. Remove the backstraps. Removing the front shoulders is very easy since they are attached by muscle. I usually use a bonesaw or hacksaw and cut in half. You could bone out if you like. Remove hindquarters and bone-out. Easy, just follow the major muscle lines. Cut steaks or roasts. Steaks are easier to cut if the meat is partially frozen. I cook the ribs so I cut them to fit my crockpot. Cut neck roast. A lot of people grind it, I use it for shredded barbecue. Grind the scraps if you have a grinder. DO NOT USE THE DEER FAT WHEN GRINDING! Use freezer bags. I have a vacuum sealer, this will keep meat a little longer than freezer wrap. If you use freezer wrap, double wrap it.
Processing gets easier with practice. Try to find yourself an old chest freezer. Chilled meat is easier to cut than flimsy meat. If you cut roasts, always rinse the cut with clean water. This will rid any fat, marrow, or tallow that gets on the meat. Wear an apron, as you will get messy.