I clean mine every couple thousand rounds. To me, best way is to take the barrel off the action amd out of the stock. Loosen the barrel band screw first, it just slips right off -then unscrew that bolt holding the stock in place it is just forward of the magazine area on the underside of the stock. Once you take the action and barrel seperate from the stock, get an Allen wrench and unscrew the two allen headed carrige bolts that hold the barrel retainer "V" block and barrel together. The barrel just slips right off then and you can clean the barrel from the breech end without buggering up the crown. The action cleans really easy this way also. In between cleanings, I generally swab out the chamber with a small rag and Q tips and wipe the gun down to get the finger prints off. The 10-22 is real simple to work on, just keep the manual handy. I always replace the bolt stop pin with a nylon aftermarket buffer pin. Pretty cheap add on and reduces wear and tear on the bolt.
I wish you good times and happy shooting with your new 10-22. I'm a big fan also of Marlins as well as CZ and Remington and Henry and Winchester 22's. A fellow can't have too many 22 rifles.