As noted, the 336 should be cleaned from the breach, and all that's required is removal of the lever screw, then the lever, then the bolt. But, do be careful not to lose the extractor. It will just fall out once the bolt is removed. Before removing the bolt note the position of the extractor, so you'll know how it should look when you replace it.

Clean the 22 the same as you'd clean any bolt action, barrel and all. 22's are dirty.

In general, I like to get some solvent into the bore and let it sit while I'm working on the rest of the gun. Then I run a wire brush through the bore and follow that with solvent patches. This is repeated until the patches look pretty clean. Then hit the bore with dry patches, and follow that with a lightly oiled patch. Before shooting next time it may be a good idea to run a dry patch to remove most of the oil.