What the others have said, but be aware that the small diameter of a .22 bore makes for a challenge. Lead cloth is on the thick side, and a .22 brush doesn't leave much room for wrapping copper strands.
Another approach is real turpentine on a tight patch of harder cotton, like 100% cotton twill. But still the same problem of the very small bore.
Perhaps a mild abrasive bore cleaner like JB compound. Problem may be that you could end up polishing the lead rather than removing it. Hard to tell for sure without a bore scope. Lead fouling is nasty because it's hard to see.
Finally, figure out what caused the leading in the first place, and try to prevent it in the future.
Good luck.
