That video is disabled. Okay ... caution, this is a bit graphic.
You need a long stick like a broom handle or an axe handle. Prepare the ground by making a small trench to lay the chicken's neck in, about 4" long by an inch or two wide. Lay the bird breast side down on the ground, grasping it by the feet. If the chicken is hysterical, calm it by hypnotizing it first (see below). Lay the stick across its neck just behind its head. Place one foot on the stick beside the bird's head. Place your other foot on the stick on the other side of the bird's head. Step down firmly on the stick and pull up sharply and rapidly on the bird's feet, pulling up to about your chest. This should instantly decapitate the bird.
There will be blood and flapping for about a minute, but the bird is dead. It's a good idea to have an empty bucket on hand to put the bird in immediately, and you may wish to have a plastic trash bag in it as well.
If you do not want blood, pull up sharply but not as high, perhaps to your waist, and you will feel a snap as the neck breaks. There will still be flapping but no blood, but the bird is just as dead, as the nerves will have been severed. The flapping is due to nerves firing. If you find it distressing, placing the bird in a bucket can help. This is the fastest, easiest and most humane way of dispatching a chicken that I have found.
*Hypnotizing a chicken. Place the chicken's head on the ground. With a stick or your finger, draw a line on the ground in front of the chicken, starting right in front of its beak and pulling slowly away from the chicken for six to twelve inches. The chicken will calm down and be still. I have no idea why this works, only that it does. Chickens are a bit weird, okay?