When I grill chicken I either grill a whole chicken cut into pieces or I grill skinless, boneless chicken breasts. Each requires a little different strategy when cooking. The challenge to skin-on chicken is the flare-ups caused by dripping chicken fat. Skinless chicken will dry out easy and can quickly lose flavor.
When grilling chicken pieces with the skin on the first thing to do to avoid flare-ups is to not pierce the meat. This will allow the fat to run out quickly and catch fire. Always use tongs when grilling chicken. The second thing to do is to keep the chicken moving. Try to save space on the grill to move chicken pieces out of the fire. If worse comes to worse move the chicken to the upper rack or off the grill completely for a minute to let the fires die down. Uncontrolled flare-ups are the number one problem when grilling chicken. Chicken should be cooked at a medium temperature, not at the highest temperature setting on your grill.
If flare-ups are a problem and you don't want to have to stand in front of the grill for the whole cooking time you can always opt for indirect cooking. This can be accomplished in a couple of ways. If you are using a charcoal grill put the coals in a ring around the edges of the grill and cook the chicken over the part where they are not any coals. If you are using a gas grill then you either have a grill with dual burners or you don't. If you have the dual burners, preheat both sides and turn one side off when you put the chicken on. If you don't have the dual burners then make a pan out of a piece of aluminum foil and place if over the briquettes (or lava rocks or whatever) and under the cooking grate. Place the chicken directly over the foil. The foil will catch the fat and keep it from reaching the burners. The downside of indirect cooking is that it will take longer. Add about 10%-20% more cooking time, but keep a close eye on things.
When cooking skinless chicken always make sure you put something on the chicken to keep it from drying out. This can be a thin coat of cooking oil or a marinade. Also, flatten chicken breasts out before you grill them. If you don't then the thin parts will dry out before the thick parts get cooking. Pounding the chicken with a kitchen mallet will make the cooking more even.
There is a third method for grilling chicken of course and that's to use a rotisserie. Grilling chicken on a rotisserie will take some time but its well worth the wait. To keep the chicken from burning and to avoid flare-ups put a piece of foil directly beneath where the chicken will spin to catch the escaping fats. You will also want to turn the grill down as low as you can to make sure the inside has a chance to cook before the outside gets overdone.