To clean a mildly dirty freewheel, I remove the cogs from the body, clean the cogs, and replace. Takes 10 minutes, a chain whip and a bench vise.
To clean a really nasty freewheel I remove the cogs and then disassemble the body. The body lockring is usually left-hand threaded, but not on all freewheels so check by trying left first, then right. Some (Suntour Winners?) actually have the thread direction marked on the lock ring. I clean all the individual parts in mineral spirits and then reassemble with a lightweight grease. I generally reuse the old bearings because FW bearings (and races) never really carry a load when in use (that's where the "free" comes from, as in no-load pedaling or coasting) and seem to wear at a rate much slower than the teeth on the cogs. Takes me about 20 minutes and only requires a chain whip, bench vise, the proper FW removing tool, and a pin tool (usually used to set the BB adjustable cup). I just sold two Lambert FWs on eBay that I rebuilt. They are way smooth now.