I've been trying to make grilled cheese on the grill for a while now. While at first this sounds funny, but I believe that it is 100% possible to do. I just have no idea even how to approach this meal. Also, please keep in mind that this is my first post here so apologies if this has been asked before or seems like a joke.
Prepare your sandwich [buttered/dry - cheese - dry/buttered] and wrap in aluminum foil. Place on the grill (or directly on the coals), turning to toast on both sides of the bread. Timing is going to be a trial and error lesson. My thought is 2-4 minutes per side, but this will be largely dependant on the heat of the coals.
As noted by GdD, "grill" means one thing in American English and another in British English. This answer uses the translations provided in this StackExchange post, and focuses on the American English definition.
Grilled (American English) a.k.a. Barbecued (British English)
This section uses the American English definition of "grill" unless otherwise noted.
There seem to be two approaches to attempting cook a grilled cheese sandwich on a grill/barbecue.
Foil-Wrapped for Entire Process
One approach wraps the sandwich in foil before cooking. elbrant's answer describes the general procedure.
Direct Contact with Grill
Another approach, with a detail example described by the Fix.com blog does not wrap the sandwich at any step. Direct contact with the grill is used throughout. A selection of steps is included below; the link also includes instructions on how to create three zones with different heat characteristics when using gas or charcoal grills.
4) While the grill heats up, grate the cheese. About 1 cup of shredded cheese should be perfect for one sandwich. Brush the bread with melted butter or olive oil.
5) Place the bread in the medium zone of your grill for approximately 2 to 3 minutes or until the bread is lightly toasted.
6) Evenly distribute the shredded cheese on top of one slice of bread. Position the second slice of bread on top of the cheese. Place the sandwich on the grill over the medium to high heat section for approximately 2 to 3 minutes or until the cheese starts to bubble inside the sandwich.
7) With a metal spatula, move the sandwich to the low heat zone for an additional 3 minutes.
Bonus: Alton Brown's Hybrid Approach
Alton Brown describes a somewhat labor-intensive procedure that involves both stages where ingredients are cooked directly on the grill and stages where the whole sandwich is wrapped in foil:
Prepare a charcoal grill by starting a two-zone fire (charcoal piled under half of the grill, with no charcoal along the bottom of the other half) and wait until the charcoal is "hot and ashy"
Melt cheese in an appropriate container over indirect heat on the grill (The recipe suggests creating containers by using grill spatulas and aluminium foil). This may take 6-9 minutes
Cover both sides of two slices of bread with fat (butter or oil) before grilling them over direct heat, 1-2 minutes per side.
Assemble sandwich; prepare enough foil to wrap the sandwich, place a piece of bread on the foil, scrape the cheese out of the container onto the bread, top with the other piece of bread, then tightly wrap the sandwich
Grill the foil-wrapped sandwich over indirect heat for another 1-2 minutes
Grilled (British English) a.k.a Broiled (American English)
This section uses the British English definition of "grill" unless otherwise noted.
bob1 pointed out the Welsh rarebit in a comment. Two pieces stacked together may form an acceptable sandwich, especially if both sides of each piece of bread are grilled to the desired consistency before applying cheese and grilling again.
If you are curious why a sandwich cooked on a griddle is known as a "grilled" cheese sandwich in the United States, you may want to look into the history of the cheese dream, an American open-faced sandwich grilled in the British English sense ("broiled" in American English).