Asking for the best way and tool to do about anything will get you a ton of different responses. For a general duty pelting knife I like one with a big easy to hold handle and a small blade. Blade needs to to be about 2 1/2 long and 3/4 wide at the base. Razor sharp is best. I make mine. I make the handle from oak. Shape the blade with a dremell tool.
I like coyote trapping best. Here in Kansas we have a lot of them. Any 22 you can carry easy makes a good trapping gun. My personal favorite is a single action revolver. After you pelt or skin your coyote, toss the hide in a washing machine. Some loose hair will come out but it wont matter. Use cold water and no soap. Put fabric softener in the rinse cycle. Woolite is real good. Makes the hair poofy along with being nice and clean. Appearence is everything when you sell fur. Toss your Skins in a dryer next with the heat turned off or down low. 60 minutes is long enough. Hang it by the nose awhile to finish drying. You can sell your pelt just like this or stretch and dry it for best money. You will need to freeze the skin till you sell it if you don't flesh and stretch it. Buy a book on fur handling.
Some women (all women?
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) won't let you use their washing mashine for coyotes. Freeze them till you get a load, about 4 coyotes, thaw and go to the coin operated laundry mat. Bring a roll of paper towels to wipe out the mashines till you get your own at auction, yard sale etc. Its best not to get noticed. Or you get a five gall bucket. Put in a pelt. Fill with water and using a plumbers helper just swoosh up and down down awhile. Empty the bucket wring out the pelt. Repeat with fabric softener in the water.
Dont fool with coyotes. Just shoot them. Blood won't hurt your remake or your pelt as washing like I described above will make you extra money. It should be done even with a, dead from a snare, yote. If your worried about the blood on the ground move your trap 10 feet.