The best and most humane way that you can destroy a raccoon that you have captured in a cage is with a carbon dioxide chamber. These are not easy to get hold of, as you can probably imagine, and they are designed to be used by licensed and insured professionals. A wildlife rehabilitator will be able to step in if you have already caught the raccoon, although there is probably a charge for this service, albeit less than what the full raccoon removal service will set you back.
Also read about:
How to trap a raccoon
Raccoon trapping tips
How to kill a raccoon
When you don't have the option of a carbon dioxide chamber (who does?), you could always look at shooting the animal, but you must make sure that you can kill it in one shot. Otherwise, the death is not going to be quite as humane as you may have hoped. You will need to look into the potential legal repercussions of using a firearm in your specific area. It is obviously not recommended to use raccoons to shoot raccoons in heavily populated areas, even in your own back yard. In more rural spots, however, this might be the perfect solution.
Poison, drowning, and kill traps aren't always as humane as you would have hoped. Kill traps can have some effectiveness when they are properly set and baited, and also if they are the right size to accommodate the animal. Drowning and poison, on the other hand, should never be used as a way to kill off raccoons in a cage.
If you are unable to find a way of destroying a raccoon in cage, or you feel as if your heart just isn't it, do not be afraid to call in the professionals. Pest removal experts and wildlife rehabilitators may be able to come up with a slightly different solution than to kill the creature, in some cases.
Read about keeping raccoons away
Go back to the Raccoon Removal
page, or learn tips to do it yourself with my How to Get Rid of Raccoons