There's lots of ways to clean a barrel. One of the cheaper ways is to use thick fishing line or weed wacker line with the end melted flat and some cleaning patches wet with cleaner on them like hoppes, goo gone, ballistol, CLP etc. until the patches come clean.
You should always pull from the breach to the muzzle. Cleaning from the muzzle end can damage the crown.
I wouldn't recommend using a metal bore cleaning kits like what is used on firearms that you find at Canadian Tire. There is more chance of damaging the barrel if you use this type of cleaner. If you can find a nylon airgun pull-through kit that would work as well.
There are lots of videos on youtube that show the weed wacker line method:
https://www.youtube.com/watch?v=oOIvgC6SJbw https://www.youtube.com/watch?v=5xu6ZnX7q4M