What you'll need to do the job:
a spray can of solvent cleaner (e.g. brake cleaner and/or WD-40), a clean soft cloth, a spray can of chain lube with straw attachment, plenty of rags.

To clean the chain, the rear wheel of the bike has to be off the ground so it can spin freely. To do this, put the bike in neutral up on its centrestand. If the bike doesn't have a centrestand or a rear-wheel stand is not available and the bike has a short kickstand, a friend can hold the bike up on the kickstand. To do this, stand on the same side of the bike as the kickstand. Grab hold of the left handgrip with your left hand, and put your right hand across the middle of the bike and grab hold just underneath either the gas tank or the seat. Pull the bike toward you so it is balanced on the front wheel and the kickstand with some of the weight resting on your right leg.

Once the bike is situated spray the bottom of chain with brake cleaner while turning the wheel, completely soaking it. Then with a soft cloth in hand hold it underneath the chain right about where it meets the rear sprocket and turn the wheel (just like it would turn while riding) to soak up all the grease and gunk until the chain is clean and dry. It is preferable to do this with a cloth because a brush could push dirt into the rollers, thus shortening the chain's life. Also, even if it's tempting, don't clean with water because that promotes rust, again shortening the life of the chain. Once the chain is dry, repeat the same steps with WD-40 to get rid of dirt and cleaning solvents completely.

Once the chain is completely clean, it is essential to lube it up. Doing this keeps the chain pliable, which if it loses pliability, requires the engine to work the chain harder thus losing horsepower.