I had the same problem on my 91 K1500. The most common reason rear lights of any kind will fail is rotted wires. They run through wire shroud from front to back along the inside of the frame to the left tail light. Running this way leaves them exposed to the elements.
Check along the run for rotted, broken or degraded wire. Most spots would be at anchor points, junctions, etc. If you do find a faulty wire, just run a new one from the engine bay to the back and splice it in. Its a cheap fix and takes WAY less time.
If it was me i would clip the wire at the tail light harness, about 4 inches back, then move to the engine bay and look for the same wire there. it will most likely be beside your brake fluid assembly, in a cluster, entering the firewall. Find the wire and clip it about 4 inches back too. Solder and shrink tube the new wire to these pigtails and give it a test.
If you are unsure of the wire color associated with brake light function, you can access wiring diagrams online, but be warned, chev/gm have been known to change color codes from time to time.
It could also be a relay, the box is located in the engine bay and the relay will be marked, but its unlikely to be the problem.
There is a switch. I think on most C/K 1500/2500 it is on the brake pedal arm or the arm hits it. It is $5-$10; you might take ALL your truck data to the auto parts store, buy a new switch, then look for the same thing around the brake arm or under the hood.
There's a fuse. Find the fusebox and look for "brake". The Owner's Manual may have clues.
Fuses usually blow for a reason. Some bulb-faults can blow a fuse. Change the brake lamp bulbs; on a vehicle this old it is time to do that. Open tailgate, there's two (rusted) Phillips screws into the taillight lens. Unscrew, the lens un-hooks on the other side. Flat rubbery thing with connector and 3 lumps. Unhook connector latch and carefully pull out. Use 1/4 nutdriver to remove 2 screws in rubbery thing. Three bulbs: they just pull out but won't want to come easy. IIRC bottom is backup and the top one is brake.
Look at the wires, because if it is not switch and not a bulb-fault, then as TBAS says it is probably wires, pinched or broken, from tail to switch to fusebox.