Keep in mind that you are cleaning an airgun and not a firearm. Use of 'strong' chemicals intended for removing primer/powder/lead residue in firearms barrels isn't necessary and could be very harmful.
For a general degreasing I use Isopropyl Alcohol in the highest concentration I can easily/inexpensively obtain. That's typically 91 to 96% found in the first aid aisle of my local drug store. It's often the recommended first step, before application of many modern high tech firearms lubricants, because there's virtually no residual film left on the metal as with some degreasing products.
I typically use Break-Free® CLP for cleaning all my airguns and firearms. For tougher jobs like cleaning the bore of a new airgun barrel I may, or may not, use tiny amounts of J-B® Non-Embedding Bore Cleaning Compound... NON-EMBEDDING. It depends on how clean I think the bore looks after using CLP. A small flashlight that can be held close to the barrel breech will help illuminate the subject.
There are hundreds of products to choose from and the general consensus is that it's a very bad idea to allow nearly all these cleaning/lubricating products to get inside the compression chamber where they could go BOOM!
All the above is based on my research, opinions and general experiences alone and I offer no guarantees. In other words, there are many 'right' products and 'right' ways to do what you have in mind to do.
If you value your fingers be very, very careful when cleaning the bore of that airgun! I suggest removing the action from the stock. After cocking the rifle the anti-beartrap mechanism can be secured so that it's highly unlikely to be accidentally released while the barrel is being cleaned. Better yet, buy or build a spring compressor, to safely release the re-load on the mainspring, and then it won't be necessary to be performing maintenance on a cocked airgun.
Finally, I've very sparingly used Shooter's Choice FP-10 to lube pellets, since the product became available, with no ill effects that I've been able to see.
Here are some results of a GOOGLE search for "oiling pellets"...
1.)
http://www.airhog.com/airblog/2006/10/oiling-pellets-is-not-straightforward.html
2.)
http://www.airhog.com/airblog/archive/2006_01_01_airhog-airblog_archive.html
3.)
http://home.hiwaay.net/~ispellan/S200.html and scroll down the page to the text heading "Barrel Cleaning and Dirty Pellets."
4.)
http://www.pyramydair.com/blog/2006/04/can-you-keep-clean-barrel-with-crosman.html
Cheers,
GH