
If they're young chicks you may not need a hatchet or axe, a good sharp pair of scissors will do. Sometimes it helps to wrap the chick in a couple of paper towels, leaving the neck exposed, so you don't have to see it. If the chicks are a bit older, you could use a meat cleaver instead of an axe.
Sometimes it helps to have an impartial 3rd party do the deed, a friend or neighbor that you can trade the sad burden of dispatching each other's ill or injured animals, when necessary. The kindest thing you can do, the final favor for the animal, is to give it a sure quick end. When you're not so emotionally attatched to the animal you can concentrate better on doing the job quickly & well.