These chains with a master link are much easier to remove than modern "linkless" varieties. With the chain off the sprockets you can soak it in a kerosene and scrub it down with a brush, all in the messy solution of kerosene. Then you can dip a rag in the solution and wipe down the sprockets to clean them. Remove the chain from the kerosene dip/wash and dry it off with a rag or really get messy by blowing it dry with an air hose. Then you can coil it up and spray it all over with a good chain lub, preferably the wax-based ones that tend to cling better. Lubing it off the sprockets is much easier and thorough than on. Re-install on the sprockets, adjust tension, and you're set to go. Make sure the master link clip is seated and in the correct direction before you are finished.
Another old-school technique before the days of spray cans of chain lub, was to heat the chain up in the oven in a pan or on the stove in a pot with good axle grease on it. (My mom balled me out for doing that one time.)
It's a dirty, messy job - however you do it...but its gotta be done.