What you're describing is a horizontal smoker, not a grill. AS you say, smokers come horizontal or vertical. The drum-shaped smokers are usually horizontal. The idea is to build a smoky wood fire in a box attached to one end, as you described, and attach a smoke stack to the other end, so the chimney draw will draw the hot smoke across your food. The heat is usually not over 200 degrees, and it's the slow heat and smoke that cook and cure the food over a long period -- Texans smoke a brisket in a horizontal smoker for 10-12 hours.
If you want a grill, forget the firebox and side-mounted smoke stack. lay the drum on it's side, cut it roughly in half (I'd go about 2/3 on the bottom or a little less, maybe 5/8, and the rest for the lid). Put hinges on it and a handle to lift it. Some folks weld arms on the back of the lid and attach a counterweight to help balance the lid when lifting it, and if you position the arms correctly, they'll also act as a stop so the lid doesn't open too far. Weld some tabs to the inside to set your grates on.
Build a charcoal fire in the bottom of the thing and grill away. If you cut some holes in the bottom, you can make a sliding flap to cover and uncover the holes and vary the amount of air that gets in -- that will help control the temperature. You should also have some holes or a sloppy fit in the lid so the heat can exit and draw across the food.
It won't last more than a season or two, unless you also build a second grate in the bottom to hold the charcoal away from the skin of the barrel. A trap door in the bottom so you can brush out the ash would also be neat.
Seems like a lot of trouble, however, when you can get a basic Weber Kettle grill for something like $79, porcelain coated steel, all the grates, engineered openings in the lid with a butterfly closure, and what I call the "propeller" in the bottom that opens and closes the air holes. A Weber charcoal kettle will outcook just about any charcoal grill in existence, and I've seen them last years and years with a little care.