Bike chains need to be cleaned and lubed every once in a while to extend their life. This is especially true after riding in the rain or mud. The chain may develop surface rust which is no big deal unless you leave it there. But cleaning, degreasing and lubing a bike chain is easy and takes like 10 minutes.
Stuff you will need include Park Tools Cyclone Chain Cleaner (chain scrubber tool), citrus degreaser, a couple rags, and chain lube. You can buy Park Tools or some other brands of degreaser at a bike shop, but it is much cheaper to go to Home Depot and buy a gallon sized bottle of citrus degreaser, and it works the same.
First, shift the bike into the lowest gear. Prop the bike up against a post or wall, or mount it in a bike stand if you have one. This is going to be messy, so do it outside, or put some sheets down. Open the Park Tools chain scrubber and fill it with citrus degreaser up to the fill line. Clamp the chain scrubber onto the lower part of the chain, where it goes between the chain rings and the derailleur. While holding the handle of the chain scrubber with one hand, rotate the pedals in reverse (backpedal). Allow the chain to go through the chain scrubber. Dirty degreaser will probably be splattering around at this point. Rotate the cranks at least 30 times.
Remove the chain scrubber from the chain. Discard the dirty degreaser. If the chain is still dirty or rusty, refill the chain scrubber tool with fresh degreaser and repeat the above process. If the chain is very dirty, or may take two or more repetitions.
When the chain looks clean, hold a clean rag around the bottom part of the chain where the chain scrubber was attached, and backpedal. You may have to reposition the rag to soak up as much of the degreaser from the chain as possible.
Now that the chain is clean and dry, it’s time to lube it. Take marker pen, and make a mark on the side of one of the chain links. Starting at the marked link, drip a single drop of chain lube on each link of the chain (on the roller/rivet part), while slowly backpedaling the cranks. Continue to do this until you reach the marked link. Now run the chain through a clean rag to remove any excess chain lube.
Congratulations, you are done!