Here's a good tip I found for cleaning a chain (assuming you have a speed link). Take a 2 litre plastic bottle of coke & drink it. Remove the chain from the bike, drop it into the bottle & add a little white spirit. Shake the bottle up, pour out the filthy white spirit & repeat until the chain is clean. Put it back on the bike & re-lube. This is much easier & more thorough than doing it on the bike.

By the way - after you lube the chain, always wipe it really thoroughly all over (until the cloth comes away clean). The chain really only needs lube on the inside - on the outside it gets all over you & attacts grit (which really wears the chain). We had a customer go through a chain in 300 miles through running it dirty & gritty through a UK winter.