Picking the large primary wing feathers should be done first, or at least early on in processing.  They can be difficult to pick.  Grasp each feather and twist slightly until it breaks free, then pull out.  The smaller wing coverts near the tip of the wing are annoyingly difficult to pick.  I've used small pliers to grasp the feathers, or you can try pulling sharply back and down along parallel to the edge of the wing. 

Rough pick the rest of the feathers from the body.  Be careful when picking a bird with a lot of fat under the skin, as it tears more easily that a lean bird. 

Be sure to get the small feathers in the "wingpits" and the upper thigh where it joins the body. 

When you have a good rough picking done, give the bird a rinse to clean off the small loose feathers.  Give the table a quick spritz, too, to provide a clean work surface. 

A clean bird, ready for further processing. At this stage, go over the bird carefully for the small undeveloped pinfeathers just under the skin.  Press or pop these out, as they can be unappetizing in appearance.  Dark feathered birds can also leave a bit of feather pigment in the follicles, so scrape or pop this material out.

Bend the leg straight, and cut between the joints.  You shouldn't have to exert much pressure if you have a sharp knife.  Just keep tension while bending the leg backward, and the bones will separate easily. 

Cut the tendons between the bones, not through the bone itself.  You don't have to worry about a bit of the scaled skin remaining on the drumsticks, as it won't affect the bird in cooking.  It can also be peeled off after the bird has aged in the refrigerator, or before cooking.

Cut off the head with a sharp knife above the cuts to bleed out the bird.  If you're doing a chicken, you may be able to cut right through the bone cleanly.  Don't be overeager to dull your knife - the head will pull off if you bend it backwards.  You can, though, use a cleaver.

If you are butchering a turkey, cut the skin off below the caruncled surface of the head and wattles, and cut the muscle tissue around the bone below the head.  Pull the head backwards and break it free. 

After removing the head and shanks, and giving it a thorough going over for pinfeathers, give the bird and the work surface a good rinse.

The compost bucket with the feathers, head, and feet.  All this material can be composted, and adds valuable nitrogen to your finished compost.  Put all inedible offal in this bucket - intestines, lungs, etc.