For a really clean chain use a 50/50 mix of gas/diesel, or carburetor cleaner like Gunk. You don't need to take the chain off, just take off the wheel so you don't get solvent in the hub or on the tire. I use a "sleeping hub" to hold the chain. A screwdriver in the dropouts can work too, unless you've got vertical ones. Slop the 50/50 on with a paintbrush, or just spray on the Gunk. Either way, use plenty. The stuff can damage some paints, so test an inconspicuous area first. I wouldn't do it at all with a carbon bike. Let it soak for a few minutes and then go at it with a brush. Then rinse everything off with the hose, allow to dry, and oil.

There's always somebody who responds by saying that this technique will ruin your chain, bearings, and/or hairdo. I won't argue except to point out that pro team mechanics used this technique from the '40s until carbon frames took over in the '90s.

For those who worry, though, WD-40 does a surprisingly good job cleaning AND lubing. Buy WD-40 in the big spray bottle, which can be refilled from the surprisingly cheap gallon can. Spray the hell out of the chain. Again a good idea to take the rear wheel off first. Once the chain is soaked, use a paintbrush or similar to get into the rollers and links and loosen up the gunk. Then spray the hell out of it again to wash away said gunk. Done weekly, this keeps the chain clean. If you let the gunk build up, it takes repeated efforts. WD-40 isn't the most effective solvent in the world, although it does a better job than most of the expensive citrus whatever enviro-friendly crap they'll sell you at the bike boutique. Anyway, after the last application of WD-40 you'll want to give the chain a thorough wipe down with a rag, a young girl's panties, grandma's used diaper, or whatever.

Dry lube, including wax and White Lightning? IMO it's not worth much, especially on a fixed gear. It does keep the chain clean, but you have to re-apply it every 50 miles or it'll squeak. If I wanted a clean, squeaky chain I'd just leave it dry.

Somebody asked for a good chain that will last. IMO this rules out anything from KMC. The Izumi V is the best chain on the planet. If it doesn't last you're either an animal or just not taking enough care of it.