Step 1: Acquire a catfish
1. Catch your own. Juglines/trotlines are a good way to catch these bottom-feeders; just make sure you're not fishing in a polluted or heavy-metal laden waterway. Each US state has its own report on this subject.
2. Get one from friends who fish, but usually catch-and-release because they're scared of cleaning fish. Bass fishermen are great for this; my uncle only keeps his fish when I come to visit.
3. Buy a live one. If you have a good Chinatown, farmers' market, or live near a fish farm, this may be easy. It's sad that you're missing out on a nice day at the lake or the river, but at least you have a live fish. This ~2.5lb fish came from the Old Oakland Farmers' Market. I was in line behind a bunch of tiny asian grandmothers waiting to buy live catfish.
4. Buy a dead one. This is a total cop-out, but sometimes the only option. Fishmongers and grocery stores standardly sell fillets, but good ones may sell you whole fish for waaaaay cheaper. Sometimes they're even pre-skinned for you. Note that "basa" is catfish from Thailand forcibly renamed by the US government.
Step 2: Kill it
To pith your catfish, position the pointy part of a big knife directly above its brain and spinal cord, then quickly push down into the brain cavity. This is about as quick and painless as euthanasia gets.
Step 3: Cut around the head
This creates an edge from which to start skinning.
Step 4: Remove fins
This farm-raised catfish has small soft fins; the average lake or channel cat has much larger, spinier fins capable of drawing blood and slicing through inflatable boats. (I've had to swim my trot line and mostly deflated boat back from the center of a pond before.) Make sure to remove these more dangerous fins as soon as possible to minimize danger in handling, and beware of the spines while landing your fish!
Step 5: Remove skin
My catfish's skin would not stand up to this sort of abuse, so I used the tried-and-true slow method of skin removal: fingers (or tweezers or pliers if you prefer) and a sharp knife. Just work your way from the head back and the top down. Keep your knife as close to the skin as possible; the sharper your knife the easier this will be. (Thanks, Tim!)
Hint: instead of trying to cut the skin off the catfish, think of it as cutting the fish off of its ski; you want to leave as much flesh as possible on the body of the fish.
Step 6: Remove head and guts
Once the head is gone, cut a slit along the belly of the fish all the way back to the anal fin at the rear of the stomach cavity. Scoop out all of the suspicious bits and pile them with the head. Try not to puncture the guts, as this will make more of a stinky mess. Give the fish a rinse to make sure you got it all.
What to do with the head and guts? You can bury them under some deserving plants in your yard, though make sure to dig deep enough to keep racoons from unearthing them. If you live near an appropriate ocean, you can use the heads to bait your crab traps. Minus the bitter-tasting gills and washed free of residual guts, the heads can be used in making fish stock. Freeze the (cleaned and prepared) head in a dated bag until you're next ready to make stock.
Step 7: Understand fish morphology
Step 8: Filleting- dorsal cut
Step 9: Filleting- ventral cut
When you reach the ribs you'll need to trim carefully to avoid leaving ribs in your fillet. Think carefully: what are you planning to do with this piece of fish? If you're going to cut it into stir-fry nuggets it's OK to be faster and a bit messy in the rib area, while if you're going to use the fillets whole you should be more careful to keep them intact. This is most easily done by working from the backbone down, skimming along the ribs to remove as much flesh as possible.
Step 10: Filleting- second fillet
Step 11: Trim and freeze carcass
Bag the carcass and put it in the freezer to use the next time I make fish stock. (Mmmmm, gumbo.) Label it with the month and year to avoid confusion.
Step 12: Clean up your mess
If you were smart enough to do this outside, bury the scraps and hose off the grass. Bonus: use the hose to spray any feral cats who have ventured too close during this operation.
Wash your hands thoroughly before proceeding to cook your dinner.
Step 13: Cook your catfish
Catfish can stand up to extremely strong flavors, so are often used in very spicy creole or east asian dishes. It's good baked, steamed, deep-fried, pan-fried, stir-fried, in gumbo or soup, and in fish tacos. This is a picture of a stir-fry in progress; so far we've got onions, jalapenos, and sweet potatoes. Unfortunately my camera ran out of battery at that point, so it won't get its own instructable.
I'll put up linked projects as I make more catfish-based dinners, and would love to see what everyone else does with their catfish.