I reckon the worst bit about cleaning the bike is the chain, but it needs to be done regularly to prevent premature wear and tear.
I think the best invention is the master/quick link. They are very easy to install and the link itself does not cost very much. If you are not confident breaking the chain using a chain breaker take it down to the LBS and they will install the link in no time.
With this link installed you can simply squeeze the link and the links will come apart allowing you to remove the chain from the bike. The next step is to degrease the chain. I wouldn't waste your money on expensive bike specific degreasers when you can get much cheaper in the auto shop for exactly the same kind of product. I put my chain in a container and pour the degreaser on top and leave it for about 2 minutes to allow the degreaser to work its magic. Then I come back with a old toothbrush and scrub the chain removing any visible grease/grit. Once this is done you want to remove the degreasing agent, so a good rinse with water does the job. You then need to lay the chain out to dry otherwise it will rust. I put my chain on a bit of newspaper in the sun or in front of the heater.
While it is drying you can clean the other bits of the drivetrain, including the cluster, jockey wheels and chain rings. If your cluster is full of crap you can use a bit of degreaser on the toothbrush to remove it with a rag. After chain is dry put it back on bike and add chain lube.