To achieve this goal you want to sear the chicken pieces first over a hot fire. Then move the pieces to a low fire to finish off. If you are using a charcoal grill you will want to build a two-level fire. Typically this means a double layer of coals on one side of the grill and a single layer of coals on the other side. With a gas grill you will want to heat the grill on high, add the chicken to sear and then lower the flame to finish it off. With the charcoal grill, place the breasts over the coolest part of the grill to keep them tender and moist. On a gas grill turn one side down as low as you can and keep the other side a little warmer. Place the breast on the cooler side and everything else over the warmer part.
Some people find that over a charcoal fire the chicken can develop an over powering smoky flavor. If you wish to keep the smoke flavor to a minimum cook the chicken with the lid off. Because gas burns cleaner you can lower the lid on gas grills. Of course with the lid off the chicken will take longer to cook. The different parts will take different amounts of time so you need to think this through before you start cooking.
The searing portion is done when all the pieces show good grill marks and the meat has just started to turn white. The skin should start to crisp and the fat will just be beginning to drain. Watch the chicken closely to avoid flare-ups and burning. From here you will need to move the pieces to cooler parts of the grill. Whole leg and thigh parts should take an additional 16 to 20 minutes to cook, while thighs or legs separated should take about 12 to 16 minutes. The breast should be cooked about 10 minutes then turned and grilled about 5 minutes more. On gas grills with the lid down the pieces should take just a minute or two less.
Of course times will vary with equipment, outdoor temperature, wind, stars, etc., so you need to keep a very close eye on everything and use a good instant read thermometer to check for doneness. Remember that chicken is cooked when it reaches an internal temperature of 160 degrees F. (71 degrees Celsius). Fortunately this also happens to be the best temperature for flavor as well.