I can't say I would suggest doing it that way, but if you absolutely want to, there is a slight trick and we call it "ringing" at least when it is a game bird.
If you were looking the animal straight on, facing it, and it's head and neck was stretched toward you, grab the bird like you were shaking someone's hand. This means when you close your hand, your fingers are under its chin and the head will be in your palm. This part is important, grip firmly but also pinch it with your thumb and the side of your index finger. This is going to force the vertebrae to separate during the ringing.
At the same time that you start pinching, move your hand in a circle (pivoting from the elbow) really fast like you are winding up a towel to snap someone in the locker room, or like that 'woop-woop' motion folks used to do on the Arsenio Hall show but down low by your waist (hopefully that hits home with someone). The spinning action isn't like you would spin a sling or lasso with the body stretched way out from your hand. Try it with a rope first to get the action down. You should be able to spin the rope without it touching you.
When you do this with the chicken, keeping a firm pinch on the neck, it will wind up that neck like the locker room towel and you will feel the vertebrae separate and only flesh will be between your thumb and finger. It's a very obvious transition from together to apart and you can stop spinning and let go of the chicken to let it do its death throws or you can cut the neck to bleed it.
The reason I don't like this method is that it makes the neck messy and doesn't get the blood out, which will affect the quality of the meat. If you only do a chicken here and there, then maybe it's not such a big deal to you.
If you have any questions on this feel free to PM me and I can probably put together a little video and send it. Animal welfare at slaughter is a very important thing to me.