As has been said here, let the meat hang for a few days. One of the big culprits that contributes to toughness is rigor mortis. That sets in within hours of death. It is a physical process that causes the muscle cells to contract and then you're trying to chew a real wad!
So, hang your animal. Skin it as soon after death as you can. For most things there's a bunch of hair that inevitably gets on the meat during skinning. Ever see a wallow? Or smell one? You don't want that on your meat, so after skinning grab your propane torch, but on the spreader adaptor and singe the meat surface thoroughly; you'll crisp up all the loose hairs. Take a clean rag and a 50:50 vinegar:water mix and wipe down the meat surface.
Put the meat into good quality, clean, cotton game bags. Not those cheese-cloth abortions, but a tight weave cloth; you'd be surprised how small a hole it takes for a fly to get its ovispositor into. If it is a big animal, hang it in quarters. With a big animal, you're going to have loose meat; dump that into its own separate game bag. With a smaller animal, like an antelope, you can likely fit it whole into a game bag. Hang it for a minimum of 4 days to let the rigor pass.
Hang it in the shade. Even if its warm, hang it for the full 4 days. On a few occasions when I've had to hang the quarters in the field, to make sure the sun didn't shine on it, I made a "tent" for each bag out of fir branches. If the bags are hung close enough to each other to touch, shove a stick or two in between them as spacers to provide for air circulation.
After 4 days, butcher the animal. As someone else said, use a knife and cut out the individual muscles, don't just saw it up with a band saw. As you cut out the meat, remove all the fat and connective tissue that you can, and make sure you get it all off the bones. Beef is a really mild meat, so it is hung on the bone and with the fat and all on it so that the slow decomposition of that stuff adds flavor to the meat. You don't want that with game because the muscle already has more flavor than beef and if you ramp that up, you get the gamey flavor.
From there, it doesn't seem to matter much whether you package it right then or put it in more clean bags and hang it longer. My own preference is to give it another few days of hanging, depending on the temperatures.
Vaccuum pack it; else you'll get freezer burn. Get a quality electric grinder. Take all the meat trim (the fat and connective tissue can be boiled for the dog) and add bacon at a 3 parts of bacon to 7 parts of meat ratio. Grind that up and it is fine, fine, burger. As the meat hangs, the outside will dry a bit. You can add that trim to your burger meat and you'll never notice it. You can also take that dried trim meat and soak it in salt water in the fridge overnight to soften it a bit, but I don't think it is necessary.
When you grind meat, you want it half frozen, otherwise it will mush up on you rather than letting the grinder blade cut it cleanly. Also, keep the knife and the screen matched. That means that when you buy the grinder, you'll need to buy extra knives; one for each diameter screen you've got. If you don't, the knives won't match the face of the screen exactly, and you'll get more mushy meat.
When you cook it, don't overdo it. Unlike beef, which has a lot of interstitial fat, cervid meat doesn't have that, so it cooks differently. Frankly, game meat makes poor fodder for the grill unless you baste it well as you cook it.
Hope this helps.
