The double bucket method is a quick and fairly painless way to dispatch a terminally ill animal that doesn't require you to be intimately/violently involved in the process. Fill two 5 gallon buckets halfway with water. Place the sedated/comatose animal into a pillowcase, and place the bag into one of the buckets, then place the second bucket into/on top of the second bucket. The animal will drown in less than a minute. You can then take the animal and dispose of it through burying or burning.
We get a lot of cats dropped off at our place because they think that since we have a few acres, we can afford it. Some of them wind up very sickly and dying. I treat them as best I can with antibiotics and good food, but those that are beyond help, I don't like to see suffer. It's a shame people can't spay and neuter more consistently, but I would go absolutely broke going to the veterinarian to take care of these strays. I don't own a gun, and I can't snap their neck.... it's too gruesome for me.
It isn't a decision I make lightly, and I hate that I have to do it. In the end, it is kinder to the poor, thrown away beasts.