My guess (and how I would do it) would be to remove the bolt, clean the barrel from the chamber side if you can with Hoppe's #9 and a nylon brush. Follow with some patches until clean or repeat the brushing. I only use metal brushes for really bad fouling if the nylon don't work.
If not too difficult, I would remove the action/barrel from the stock before cleaning/lubing. Be sure to use a tight fitting screwdriver so you don't mar the heads of the screws. A wipe with a damp cloth is all the stock should need and perhaps a coating of Pledge or similar.
Wipe any excess Hoppe's from the inside of the action as needed, perhaps use an old toothbrush to scrub the machined area if grimy. Wipe dry. Clean the outside of the bolt similarly and disassemble and clean if not too difficult. I don't know if the firing pin "floats" but it should be free and not gummed up in there. Same with the extractor. If convenient I would use a spray solvent for the bolt passages and quickly blow off with compressed air. I would use any spray solvent sparingly as they can damage gun finishes.
Once clean, I'll take a patch with gun oil (I use RemOil) and run it through the bore, wipe the outside metal parts with it, then add a couple/few drops to the inside of the action, trigger bar and bolt inside/outside. I'll put a few paper towels around the action to absorb excess oil when I put it back in the hard case.
I don't always clean a gun after a range outing unless it is really dirty or fouled. A couple drops of oil here and there and maybe an oily patch through the bore is better than nothing.
PS: after cleaning and reassembly it is always a good idea to check function. A set of dummy .22s is a great idea for safety and to avoid damage to the gun. It is not desirable to dry-fire .22 guns because that can damage the firing pin and/or chamber face.
