It all depends, of course, on how hot your fire is and how big your pieces of chicken are. We tend to go for smaller pieces around Casa Silenus, hence the shorter cooking times. YMMV, obviously.
We zap them in the microwave first just long enough to get them nicely warm on the inside.
Boneless and cut in half also helps, of course.
You could try brining them beforehand. If they’re getting burned on the outside before they’re cooked through on the inside I think the advice to the OP to use lower/indirect heat, and to be sure not to not apply any sugary sauces until the very end of cooking applies - and maybe try bone-in/skin-on breasts as Bill Door suggests.
We do this with pork steaks, too. Put in a dish with a little moisture - water or marinade- cover with saran, microwave a few minutes (the saran keeps the moisture in) and finish on the grill with the BBQ sauce. You get a done inside and crispy edges.
Coals on both sides of the bird to control the chicken and then cook it standing up beer-butt style. You should be using a cooking thermometer.
Forget that - This is what you should be looking forward to. You’ll never have dry chicken again!
I recommend this as well. I use one of these, you can get something similar at Target etc. Rub the chicken with spices, stick it on this with the beer can 1/3 full (you can use water in the can, its still good). Cook on medium-low for a while (use a meat thermometer in the breast), and the result is moist, tasty chicken with very little hassle.
ETA: you “dirt cheap grill” may not have sufficient height to close the lid on a stand-up chicken. Oh, well.
Instead of regular charcoal, I’ve recently been using lump charcoal. It’s basically charred pieces of hardwood instead of artificially bindered pellets of carbon. I’ve read it burns hotter and I think it tastes better. It at least seems more natural a fuel. They sell it along with the regular charcoal.
Yep- nuke then char.
I have had consistent perfect results with a beer can cooker and indirect heat.
First off don’t use matchlight unless you like the taste of lighter fluid. Get a charcoal chimney.
Secondly cook with the lid closed. This limits the oxygen and therefore prevents flareups. No oxygen, no flareups.
OG no. Just cook it right, no need to boil it.
Winner!
I did chicken thighs twice last week on a Weber kettle. I brined one set and did a marinade on the other set. In both cases they were perfectly cooked. I prefer brined as it is juicer, but to each his own.
Here is how to brine chicken
1 cup Kosher salt
Dissolve in 1 quart hot water
Add 3 quarts of cold water to make 1 gallon.
Add chicken and place in fridge for 2-12 hours ( I like the long end of the scale, YMMV)
How to cook bone in Thighs, legs etc:
Start your charcoal, not too much. When it is ready to cook, you want a bare single layer (a few gaps is fine) When ashed over, spread and place grill on unit. If you want to add more smoke flavor place some pre-soaked wood chips on the charcoal.
Place chicken on the grill AND CLOSE THE LID)
Turn the chicken every 10 minutes. Keep the lid closed except when turning the chicken.
Chicken will be done in 30-40 minutes.*
If by chance you didn’t get enough heat into the chicken and it isn’t quite done, 30 seconds to 1 minute in the microwave will fix that.
Bone in breasts will cook the same way, but might take a few minutes longer.
*time depends on what temp the bird was when you put it on, and the temp of your grill.
Start watching America’s Test Kitchen on PBS on a regular basis. Subscribe to their magazine Cook’s Illustrated as was mentioned earlier in this thread. You will learn the secrets of barbecued chicken. 
Just by coincidence, I grilled chicken last night.
Start up the coals, then split them 50/50 to either side of the kettle.
Seasoned* chicken parts go in the center of the grate.
Ignore for 45 minutes.
Apply BBQ sauce, if desired.
Leave it alone for another 15 minutes.
Eat.
- Something dry - salt & pepper, lemon&herb spice mix, etc. No oily dressings.