I'm sure others have gone over this many times before, but since you are a little unsure, I'll throw in my two (or three) cents as well. I like to butcher at about 12 weeks for my delaware crosses. I sometimes confine them to a smaller cage for about a week previous to "the day" feeding corn and oats. This step is mandatory for older birds as it gives their muscles time to 'lose tone', or get out of shape. On harvest day, they are not fed, only watered. They are placed in a bleach jug cone for several minutes for them to relax and the blood to rush to their heads. Using a RAZOR SHARP knife, cut both sides of the throat quickly and deeply. You will know that you have done it right since the blood pumps out at a right angle to the neck. Exsanquination is complete in less than 30 seconds but let them drain for about 5 minutes and they will be completely empty.
Immediately set to the cleaning, dip in hot (140 degree) water and pluck. Remove the crop from the throat being careful not to spill the contents. Then carefully cut the skin around the vent just enough to free it and tightly tie a string around it to prevent poop spillage. Proceed with drawing the bird, saving the giblets. Give the whole bird a thorough flushing with cold water. Make a brine of 1/4 to 1/3 cup canning salt per gallon cold water, as many gallons as needed to completely submerge all the birds you have, and let them soak for 1 1/2 to 2 hours. This doesn't make them salty but will draw out any remaining blood.
Finally, place the birds in gallon sized zip lock bags in the FRIDGE, not the freezer, for three to four days before cooking or freezing. This is the mandatory aging process that allows the enzymes present in the muscles to start to tenderize them and bring out their best flavor. Prime beef is often aged for 2 or more weeks!
I worked in the meat cutting trade as a youth while going to school and hunted a bit back then too. I found that what was often described as a "gamey flavor" could be traced to 1) Adrenaline; from a wounded, ill, or unduly agitated animal, 2) Inadiquate cooling; from leaving the gutting and skinning of an animal for too long, or 3) Contamination; spilling of intestinal contents. Minimizing these elements, and keeping healthy, well fed animals is all that is needed for delicious, home grown meat.
I'm sure your next attempt will turn out better! We never actually fail, we only learn a new way not to do something again!
~S~