It can be done alone, if you are strong enough to hold the chicken with one hand. Some chickens are too big for some people to kill this way.
Hold the bird upside down by the feet. Get both shanks in your one hand, then grab the long feathers of each wingtip and put them in the same hand that holds the feet.
The bird will usually hold still at this point, and may even stick his neck out. Pick up the hatchet with your hand that is not holding the bird, lay the neck on a chopping block, and chop the head off. Some people stick two nails in a v-shape in the chopping block: lay the neck into the V and pull back a little on the feet, to stretch the neck out for easier chopping.
This is probably obvious, but use the hatchet in your dominant hand (right for most people, left for left-handed people) and hold the bird with the other.