The only tools needed are a cleaning rod with a ball bearing swivel, brass jag and cotton patches (or Brownell's VFG pellets), and a fine abrasive like J-B Bore Compound.
1. Always work from the breech end to avoid any chance of damaging the crown.
2. Lightly oil the patch with a low viscosity oil like Kroil or pneumatic tool oil. Not saturated and dripping, just a little. Then smear a uniform coating of J-B compound onto it.
3. Apply more strokes at the breech end and fewer as you make it toward the muzzle. A good place to start for most barrels is something like 25 strokes to ¼ of the length, 50 strokes to ½ of the length, 75 strokes to ¾, and finally 100 strokes the full length.
4. Load a new patch when you feel the pressure against the inside walls of the bore has diminished quite a bit. If using the Brownell’s VFG adapter and felt pellets, you can incrementally thread the pellet down onto the taper to bump up the pressure again a few times until it eventually shreds. If using a brass jag and a cotton patch, replacement will need to be more frequent. It works fine though…don’t feel like you have to get the VFG stuff to do a good job.
5. Do not let the patch exit the muzzle. That will develop a subtle bell mouth. No bueno. Clamping a stop block of some kind at the muzzle makes it easy to avoid.
6. Clean well with a suitable solvent and pull-through patches until they come out clean.
7. If so inclined, do a followup with a finer polish like J-B Bore Bright, Flitz, etc. I always do because it’s a small additional effort compared to setting up and dragging out my supplies. I apply about half as many strokes as with the previous compound.
8. Clean again, reassemble and test!
Lastly, here's a link to a DIY article on common barrel troubleshooting and accurizing techniques.
https://www.gatewaytoairguns.org/GTA/index.php?topic=130555