I don't necessarily remove the chain as often as once a month unless you've been riding in rain and grit. Basically, if it has been dry, your cleaning yuold include using a rag and degreaser to clean the chain (cup the chain in the rag and pedal backwards - keep rotating the part of the rag you are using until it is somewhat less black and gunky on the rag). Then lube the chain wel, allow it to sit for 30 minutes and wipe it dry with a clean rag.

Cleaning other parts of the bike is simple - you can use the degreaser or simple green or whatever, and just remove visible gunk.

Pay attention to where the cables are routed below your bottom bracket. A lot of times sports drinks leaking from bottle tops or other gunk gets in there and things won't shift as well.

You're making me feel like I should go clean my bike...