Nashbar sells a chain-keeper tool like that. It works perfectly. In general, I find that you need a few inexpensive specific tools to clean a bike easily and thoroughly.

My tips:
- take the wheels off, mount a chain keeper and mount in a stand.
- get a long plastic brush the reach in behind the crank, hit the hubs and behind the brakes. I have one designed to clean fridge coils that works perfectly.
- use a stiff-bristled short handled gong brush on the cassette and rims.
- make sure your frame has good drain holes, if not, mark the seat height, pull the post and invert the bike to drain it after washing (this is a pain, the best advice is to buy a frame with good drainage...)

Method: I do this outside (obviously).
I use two buckets: one with very diluted dish soap (one squirt in several gallons) and one with plain water.
I use two automotive sponges that are wrapped in microfiber cloth, I got them at AutoZone.
I wash the bike with the soapy water first, then the drivetrain with the same soapy sponge, then use the brushes on all the hard to reach spots. I don't use any special solvent on the chain, soapy water works great.
I scrub the brakepads and cassette with a brush. Work aginst the ratchet in the cassette and really scrub, there's always an amazing amount of dirt in there. I also use a little Park cassette-specific brush to get behind the cassette.
I dig any implanted grit out of the brakepads with an awl.
I then wash everything once more with the plain water.
I wipe everything down with a towel.
I spray the drivetrain down with WD40 to displace water followed by relubing. I also relube the brakes.
I let the lube dry/sink in a bit, wipe of the excess with a rag, remount the wheels, run the bike through the gears a few times and I'm done.

I can get this done in maybe fifteen minutes and do it only as needed. I have a brushed-finish Ti frame and use a little Lemon Pledge if I want it to shine, which is rarely.

One other tip, I use a paintbrush and odorless mineral spirits on the chain to clean it if I'm not washing the whole bike down.