I like to service my chain as part of a trip to a local park. Turn it into another excuse for a bike ride. Sitting on a picnic bench, I have the bike in front of me as I sit backwards facing away from the table. I can reach a length of the chain and work on that, then I roll the bike backwards (which rolls the entire drivetrain, chain included), then forwards a few times, repeating the process until a new length of chain is in front of me, whereby I clean it and repeat until I've got it all. Drop a chain checker on it and take a peek, too.
I have a bike stand and I could do the chain maintenance in a more straightforward/traditional fashion, but I would rather make it a part of a ride. The bottle of cleaner and rag come along in a little ziploc bag no problemo.
Some folks remove the chain, do an ultrasonic clean and then wax the chain, get it blessed by the parish priest etc. I'm not that guy. I use Rock and Roll Gold in summer and Rock and roll Extreme in the wet months. Its a combination solvent and lube. You squirt on, wipe the chain thoroughly with a rag and move to the next length of links. If I have crud on the cogs and pulleys, I scrape that off, usually (carefully!) with a screwdriver tip and a rag wipe. Front chainring gets only a wipe. No metal on metal there. A couple weeks later whether it needs it or not, I do it again. the two bikes I am riding have 2000 and almost 1000 miles on them and chains and clusters are still going strong. I can expect to get 3000 before I replace cluster and chain together, if history repeats itself.