I hang the deer head-first from a tree in the yard, and skin it. I have a water hose handy, and wash everything thoroughly as I go. First I get the tenderloin from inside the body cavity. Then I fillet out the backstraps, full length. Next I remove the front quarters, and then the back quarters. All goes into a large cooler on a bed of ice, including any trimmings. The legs are cut off at the "knee" joint, which is actually the ankle, before I remove them so they will fit in the cooler. Getting the hind quarters takes a little work, finding the ball joint and dismembering it. The ribs are sawed off, and the rest is scrapped, after taking any lean trimmings. I use a knife for everything but the ribs.
If I have to do it in the shed, I put an old tarp or shower curtain under the deer to catch drips. It can be washed off, but it's best to spread it in the sun to dry afterwards.
I live in the country, so getting rid of the carcass is no problem- I put it somewhere out on my property and sometimes set a game camera to see what comes.
I cover the meat with ice, and leave it in the cooler for a week or so, propped so the melt will drain, and add more ice as needed. When I'm ready to process, I take each section into the house and work it down at the kitchen sink. I cut steaks, roasts, and burger scraps as appropriate, and wrap the steaks and roasts tightly in two layers of freezer paper. I save the lean scraps until the end, and grind them all up into burger. I trim and discard all of the fat and most of the sinew from the burger, although my grinder will handle about anything. I normally don't add any fat, but I may next time for variety. It gets double wrapped like the rest. Double wrapping with freezer paper will keep it good for a year or more.
Cleanup in the house is with soap and water, followed by Clorox wipes, at my wife's insistence.