I'm not sure how long you have owned this Fuji. Since you don't remember ever changing the chain, if I were you, instead of just cleaning the one that is on there, I would just replace it. Some suggested chains would be the SRAM PC8xx (PC830/850/870/890), KMC (I like the X8.93 or X8.99), Shimano HG 40 or HG 50, etc. I used to use the SRAM chains but have switched to the KMC.

The SRAM and KMC chains will come with a master link; you will still need a chain tool to shorten the chain to the correct length for the initial install, but after that you can just break the chain for cleaning using the master link.

For cleaning, I personally soak the chain in mineral spirits in a coffee can, then scrub it with an old toothbrush, wipe it down and let it dry before lubing and reinstalling it. Clean the rear sprockets and chainrings, as well as the derailleur pulleys, while waiting. As mentioned, there will probably be as many ways of doing this as posts in this thread. Chain lube threads can tend to get long