1) Rear Derailleur - Nothing scientific about this step. Simply get out the toothbrush, dip it in your container of degreaser and begin scrubbing away at the rear derailleur. The main area of build up will probably be the pulleys (two small toothed rings in the derailleur). Get your rag and dry the derailleur and pulleys. Try to get in all the nooks and crannies. And make sure you wipe down the pulleys. This will be hard to dry so use some ingenuity with rag. Its doable!
2) Front Derailleur - Follow the steps as you would with the rear. You will not have pulleys to clean, but you will have to get into some tight corners. So do the best you can.

IMPORTANT: Once you have finished cleaning both derailleurs, shoot some Teflon spray on the springs and tight areas to help lubricate the system.