Rotisserie chicken is without doubt a crowd pleaser. There is no reason not to cook a rotisserie chicken on a charcoal grill if it can be rotated so that the heat is radiated from the back of the grill or if the grill itself is large enough to have two piles of burning coal either side of the grill so when the chicken drips those drips can be caught in a tray. Keep topping up the charcoal so that it continues to burn down and rake the ash from the bottom so it isn't able to blow up on to your chicken.
Kettle barbecues can also be adapted to cook rotisserie chicken though hot roasting a chiken in a kettle barbecue will render very similar results. With a chicken placed on a grill inside a small baking tray the chicken sits above it's cooking juices which evaporate and help keep the chicken moist.
Many larger gas barbecues are even supplied with the necessary rotisserie equipment. If you have a gas grill with rotisserie I recommend you refer to the instructions, if you have them. Though with a little lateral thinking you should be able to work it out. With the lid of the gas barbecue closed the heat will build up very quickly. If you have a thermometer built into your gas barbecue keep an eye on the temperature and don't let it get too high. From experience a whole chicken can cook in around 1 1/2 - 2 hours depending on it's size.
My preference for rotisserie chicken is to cook it in front of my chiminea. In the summer this is great as the chiminea heats up the patio area too and we sit and watch "chicken tv" as the chicken turns gently in front of the hot fire bubling and oozing. I know when the chicken is cooked because a leg usually drops off into the dish I have waiting below. The skin is delightfullt crisp and the meat so tender you'll be drooling as you carve it.
I made two stands from metal. The sections that make each end of the rotisserie stand bolted together giving a stable structure to which the rotisserie skewer can span. The metal stands were painted with heat resistant paint even though the metal is galvanised to give it additional protection. The metal work system is used in the electrical business for carrying cables in large factories, warehouses and even airports. If you want to build your own using these materials ask a local electical wholesaller for Uni-Srut.
Spits are available in both mains or battery powered. I prefer the mains powered type as I don't then have to worry about the batteries being charged or lasting the duration of the cook. I am fortunate to have mains close by and for that reason I acknowledge the convenience of a battery powered spit. Whichever you chose check that it is suitable for the amount of food you wish to cook and for the fire or barbecue you are going to use for heat.