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.
count the links of your chain, buy a sram pc830 off ebay (under $10, shipped), and throw the old chain in the trash.
you will need a chain tool. spend at least $10 on one. don't buy a super cheap one from asia.
if you can't get the new chain's master link to click together easily, throw it away too.
Thanks, everyone. I bought the '85 Fuji off Craigslist (back in June) from a college grad who no longer needed it. The bike is darn near in MINT condition, as it was the college kid's dad's old bike, bought new in '85. I suspect it was a "garage queen" for quite a number of years, until the kid went off to college.
Anyway, as seen in the photo attachment below, I was just outside on the patio deck, working on cleaning off the old Fuji. The Suntour LePree derailleurs are still working fine with no problems. The Suntour "Mighty-6" freewheel (14/16/19/22/26/30) is still looking good but dirty and grimy with a little bit of gunk in the cogs. The chain is all blackened and nasty and has 112 links (I counted each, and then recounted each outer link (56) to double-check myself). You're going to laugh at the photo also, because on the table in the photo is a spray bottle of (believe it or not) trombone slide oil! For you fellow musicians out there, I'm talking the good stuff... The synthetic type, water-proof, slide oil for trombones. What the heck, it's ultra lightweight oil and works incredibly well! The old rag in the photo is other topic altogether, to where you all will say,
"Butch, what is the matter with you? Are you nuts?" 
But one end of the nasty rag has gun cleaner (bore solvent) on it! Wow, did that clean the chain and cogs in the freewheel! I was just extremely careful not to get the rag near the paint on the frame!
Long story short... I will ride around for a little bit, maybe this afternoon or tomorrow or so, and then just replace the nasty old chain with a new one. Most new chains are 116 links, so I'll have to cut it down to 112 with a good quality chain tool.