This is a very personal thing and there are several way to do it. I've tried several. Use whatever method you think you can actually accomplish and do well.
I use the same method, regardless of if I am euthanizing due to illness or processing for the dinner table. I use the broomstick on the ground method. There is a good write up and illustration on this method in Storey's Guide to Raising Chickens, p. 388. However, my method has evolved over time. Because I want the chicken to die instantly, I pull a little harder and the head comes off cleanly from the neck. I know it sounds gruesome, but the act of taking the life of any animal is never going to be pleasant. The advantage of removing the head is you KNOW the chicken died instantly and thoroughly. After all is said and done, this is the goal, to allow the animal the most clean death you can give him/her. And this method is EASY to do, it takes truly no skill, there is no accuracy involved (like with the hatchet or ax method), no technique of twisting the wrist a certain way or pulling a certain direction (like in the neck wringing method) and no knife-type skills. You stand on a flat surface, step on one side of the broomstick, put the chicken's head under the broomstick, step firmly on the other side of the broomstick while immediately grabbing both legs together in your 2 hands and pull straight up. Yes, the chicken flaps his/her wings and it again, is not pleasant but it has to be done.
Then, if you are processing for the dinner table, you hold the chicken upside down inside your bucket and let the blood drain out. If you are killing the chicken because he/she is sick, you carry the chicken body and head to the hole and bury him/her. Either way, you thank the bird for his/her life, say goodbye and maybe shed a tear or two. It's never easy for me or many of us, no matter how many times we have to do it.
Guppy