Preparation is the key. Gather your tools first, then remove all hardware, handle, corners, feet, chassis, grille, etc.
Get a couple of cheap toothbrushes and a soft bristle brush. Spray your cleaner generously on one side at a time, I happen to use Armor all just fine, and gently scrub with circular motions. This will get years of dust and grime out of the crevices of the tolex while brushing the cleaner into the crevices for that crazy makes-you-stare-at-it shine. Use a toothbrush to get the smaller and hard to reach areas such as the inside corners (control panel and grille area) and ledges on the front of the cab which frame the grille and control panel. After each side, wipe off the excess cleaner, look at that paper towel, it's totally gross! But the cab looks great.
Take the grill and scrub with a dry toothbrush in circular motions to get the dust out. I hesitate to use any cleaner unless needed because it's a pain. When it is needed, I spray, scrub, then blot dry with paper towels. Don't wipe on the grill, you'll just leave behind paper towel boogers. Blotting takes a lot of time, so does using a toothbrush, but it'll work.
If the grill is jute, a natural plant fiber, I do not use Armor all, I use Doctor's Wood Cleaner to clean, and Bore Doctor to condition the jute, both are fretboard care products. After cleaning, condition the jute by applying the Bore Doctor with a sponge brush and blot it on, don't brush.
For the edges where the tolex has peeled, get contact cement, a roller, and those tiny paint brushes from the craft store. Make sure both surfaces are clean and dry, pull pack the tolex a bit, brush on contact cement to both surfaces. Then, you must allow the cement to completely cure
before you assemble. Use the roller to apply firm pressure and just like ironing your clothes, press that tolex to the wood and it'll be good as new.