1. If you're in an area that uses tags, tag your deer or other game animal immediately. The tag must remain with the deer at all times, or you risk confiscation of your deer among other nasty effects.
2. Carefully cut a circle around the anus so it's free and can be removed from within. Some folks tie it off with string to prevent its contents from tainting the meat.
3. If it's a buck, remove and discard the testicles and cut the penis free so that it can be removed by the same route as the anus.
4. Beginning close to pelvis, open the stomach cavity to the ribcage. After starting the cut, use the first two fingers of your other hand to help guide your knife... you must only cut through skin and a thin layer of meat, and miss the entrails.
5. Be careful NOT to cut past the shoulder area if you want this animal to be mounted!
6. If you want it mounted, get that cape cooled down soon to prevent hair slippage when tanning. 
7. Sever the windpipe and esophagus at the base of the skull.
8. Cut the diaphragm loose. This is the sheet of muscle that separates the stomach area from the chest cavity.
9. Allow the animal to roll on its side, and "help" the organs to come out. You'll have to pull a little, but they should be mostly free.
10. Be extremely careful in removing the bladder! You must reach up into the pelvis and pinch it shut while you cut it free with the other hand. If any urine is spilled on the meat, remove it immediately with water from a thermos or a clean cloth.
11. Clean any debris from the cavity. Any stomach contents or other substances should be removed as quickly as possible.
12. Separate the tenderloins now while they're fresh. Cloth bags are recommended for keeping these clean and allowing them to cool.
13. Get animal to cool down as soon as possible to prevent bacteria and meat spoilage. 

Note: Examine the deer for signs of old wounds. Your broadhead, a sharp piece of bullet metal, or an old broadhead (though unlikely) may be present, and could cut you if you're not careful. Also look for signs of chronic wasting disease! Please refer to http://www.cwd-info.org/ for excellent information on how to handle animals to safe guard yourself against such a disease.