Homestead Poultry Butchering: Killing the Bird
I agree, there is no "good" way to dispatch an animal, but the neck-break or a sharp axe and a steady, strong hand will be the least traumatic. The flailing around of the body afterward is disconcerting, but honestly the animal has felt none of that. I've seen people club a bird at the join of the neck and skull with the back of an axe or a hammer before, but I don't understand since you will need to behead the chicken later for eating. In your case, I am not sure if that is on the menu, so to speak. You may find that if you don't want to mutilate the body and bury it like a pet, the hammer is better than the axe, and it doesn't feel the same as wringing the bird's neck by hand. Better for those who may be squeemish.
If you tuck the bird's head under it's wing while you walk it out to the block, it will keep it calm and sedated. Once you are there, do the deed quickly so that you are not putting the bird or yourself under stress for too long. Don't procrastinate or overanalyze at the time. Just do it. There will be plenty of time afterward for you to soul-search and rationalize, but the bird's end should be as quick as possible, without a lot of drama.
If you use chickens as food, the killing process gets easier as time goes on, and you will eventually find that you learn the routine to make it go smoothly. The first time is always the toughest though.
Good luck.