Mineral spirits or Stoddard Solvent for cleaning. (I try to avoid breathing Lacquer thinner fumes whenever possible.)
Blow dry or let dry overnight.
I like PJ1 chain lube. Has a tough film suspended in a light solvent. Kinda Foams on and penetrates the rollers and links, then the solvent carrier evaporates overnite, to minimize fly off. If you get fly off, a little mineral spirits will do well for cleaning that, too.
I don't care for dry lubes on non-oring chains. I just don't believe they lubricate well. They will keep rust at bay, though, and minimize fly off. I speculate the "clean" issue dominates its choice for many.
Anyway, with the PJ1 (Black) used regularly, you don't need repeated separate cleanings of the chain, in my experience.
Cheers,