The "Deep Cleaning" process that I'll describe will take just a little bit of your time, a good gun cleaning product, I've used Hoppe's #9 for years, a good brass/bronze brush of the appropriate size, a cleaning rod, some good cotton patches, a warm work area that's protected from the mess, and some elbow grease.
This is the step-by-step process that I recommend:
Remove the barrel from the receiver.
Install a cylinder or improved cylinder* choke tube.
Spray the bore of your gun and then spray your brush very liberally until it's dripping (I use a wide mouth jar filled with Hoppe's #9 and dip my brush in it from time to time to keep it wet) and begin to “scrub” the barrel from the chamber to the muzzle end for several minutes. I recommend no less than 4-5 minutes. Really scrub it and use additional solvent on your brush and/or in the bore, as needed, to keep it very wet. DO NOT SWAB IT OR DRY IT.
After your scrub (for at least 4-5 minutes, longer is better – don't cheat on the time) let the barrel set wet, don't swab it, for at least 10-15 minutes at room temperature.
After you've let it set, scrub your wet bore again with your brush for a few minutes, use more solvent on your brush and/or in the bore to keep your brush wet and to help “wash” your barrel while you're doing this.
Now, it's time to use the cotton patches. (I usually switch to a Tynex/plastic bore brush for this operation but you can use an older brass brush.) Place a patch over the brush and swab your barrel until at least 1 comes out clean and dry.
Remove the choke tube you installed and clean the area that it occupied, including the threads, with a toothbrush or brass brush made for this purpose. Don't use the brass brush you used for cleaning the barrel. Wipe it dry with a lint free cloth and then lightly lubricate the area. You should always keep some type of choke in your barrel to prevent foreign material from fouling the threads.
Look at your barrel. It should be shiny and bright.
(If it has pitting, gouges, machine marks, etc. it may never pattern as well as you'd like and you may want to have it professionally polished. Be aware that may cost $100.00 or more. You and only you can decide if a better pattern is worth the additional cost.)
Reassemble your gun.
If you're ready to pattern test it now, leave the bore dry and install your turkey choke. If you plan to store it, you may want to put a protective coat of lubricant on the exterior and interior of the barrel. Be certain to remove all lubricant prior to shooting.
This process involves some work but it will pay dividends in most cases by helping to increase the quality of your patterns.
Lastly, be certain to dispose of all rags, newspapers, etc. in a responsible and safe manner. Some gun cleaning products are flammable and must be treated as such. Disposal may require some extraordinary caution. Some gun cleaning products are harmful if used in an unventilated area. Please read and follow all label directions and cautions on containers.
*This is done to make it easier to clean the barrel. It is sometimes difficult run a cleaning rod, brush or patches through tight turkey chokes. Ported chokes can be hard on brass brushes and sometimes "eat" cleaning patches. You should clean your turkey choke with at least as much care as you clean your barrel.