There are countless ways to cook chicken on the grill. I'll cover one of my personal favorites here, with more to come.
Beer in the Butt Chicken (aka: Beer Can Chicken)
Smoked "beer in the butt" chicken is my personal favorite. The chicken comes out tasting awesome and extremely juicy. I can't stand dry chicken, and chicken is naturally dry when cooked thoroughly. Alcohol tenderizes meat and as the beer is evaporated up through the chicken, it makes it tender, juicy, and great-tasting. Don't plan on eating the skin, though because it gets very tough during the smoking process, but this is what keeps all the moisture in the meat as well.
Make sure your beer is about half full. I like to dump some garlic powder down in the beer.
Set the chicken on the can holder. You can use a rub if you like, but you don't have to. I prefer to use a rub. Even though I don't eat the skin, the flavor still works its way into the meat during smoking.
If you don't have a can holder, you can set the chicken directly on the beer can and use the legs to balance the bird.
Stuff a piece of fat or foil in the neck hole. This keeps all of the steam inside and helps it cook better from the inside.
Cook the chicken long enough to get the internal temperature up to a safe degree for poultry before taking it off. I generally cook mine for at least 3-4 hours depending on how hot my fire is. You'll want a fire around 275-300 degrees.