Routine cleaning is outside. Spray down the drive train with Simple Green or something similar and then wash with soap (Lemon Joy) and water - NO PRESSURE and do not aim directly at openings.

Deep cleaning is done with the Park chain cleaner thingie and Park chain cleaner or the Pedro's equivalent. These cleaners is not as environmentally friendly as Simple Green but do a far better job.

Park Tool's website is great for directions and recommended tools. Ditto what yellow said.

All this is best done in your driveway on a sunny day with a friend. One washes bikes and one fixes and serves the Gin and Tonics, perferably Bombay Sapphire.