The first year I butchered a few birds I tried the knife thru the roof of the mouth into the brain. After almost sticking a knife thru the palm of my hand I decided against that. I've also tried slicing their neck. But after all that I finally went back to the old tried and true. Probably similar "broomstick" method Myrna mentioned. I hold the bird upside down until it calms down and goes limp. Then holding the bird by the feet, I lay the head on the ground and in one quick move I put the outer ball of my foot behind the neck and pull up. Unless you wnat the skin on the neck you can just pull the head right off. Thus the old saw "running around like a chicken with its head off".
Whether I skin or pluck depends a lot on how many I'm doing, whether I'm by myself or have help, how much time I have, how much chicken broth/stock I have on hand, and especially the condition of the bird. If it's full of pin feathers I definitely skin. One thing I will mention is that if you're plucking it really works best if you have one of those outside propane stoves where you can keep a big pot of water on at 150 degrees. The feathers literally come sliding off. I prefer to do about 10-20 birds at time.
Since I never cook a bird whole (unless it's a guinea or peahen) whether I skin or pluck I do the same thing to butcher. Have two 5gal buckets. One for innards and the other with some ice water. The first thing I do is remove the neck and the crop. Neck in the ice water and crop in the innards bucket. Then I remove whole tail and the vent area. That creates a nice little opening for the innards to come thru. Then I take the legs and thighs off in one piece from the back making sure to keep the oyster with the thighs. Then I slice the skin on the abdomen along the fat lines below the ribs and then thru the center to open up the gut cavity. Hold it over the 5gal bucket and they'll drop right in. I don't eat gizzards or liver so they go to somebody that wants them or the catfish in the pond. Same with the heart later on. Then I remove the back from the breast, pull the lungs out, and that's it.
From there I take it in the house, clean all the pieces up good, separate the legs from the thighs, split the breasts, and the vacuum pack it all. You can pack them by bird or by parts. Personally, I like to package them by parts. 8-10 legs in a pack. Same with thighs. Maybe more with wings. 1-2 breasts per pack. Backs and necks go together for broth and stock.
One last note, I rarely cook since it's just me so normally I'll use the crockpot, stack the piece in, cover it with water, and let it cook overnight. If I'm making chicken noodle (or rice) soup I'll use legs, thigh, and wings. If I'm wanting meat for fajitas or enchiladas or something like that I'll use breasts. I just cook it until the meat literally falls off the bones.
I know I've butchered roosters that were more than 2yrs old and cooking it in the crockpot until the meat falls off the bones is about as tender as you're gonna get. I recently read about "bone broth" and think I'll give that a try. I also read somewhere that chicken feet make for some good broth so I may try adding them next time.