Some bits of rag, hot or warm water from your escouade's kettle or marmite (lacking that, alternatives are in the text...), olive oil (a small tube, the size of a cartouche, filled with olive oil should be in your giberne), a small box of brick dust (which should be in your backpack), a bit of lard (was issued to every soldier for this purpose, and should be in your giberne or backpack), your musket's ramrod, and a tire-balle or ball puller or worm (at least the escouade's corporal should have one, but often veterans had their own).
How to do it?
With a wet piece of linen, wipe off the black powder remains and residue from bayonet, ramrod, lock and barrel. The water (or lacking that, saliva or urine) should dissolve these remains pretty easily. Make sure you don't wait too long, as dried up residue will attract humidity and this will make your musket rust...
Pour hot or warm water into the barrel, until it comes out
clean. French fusiliers would often urinate into the barrel to achieve the same
effect. The same thing was also done in battle, when, after heavy volleying, the
barrel was clogged up with black powder residue.
An alternative is to plug the touch hole (with a bit of wood, for example), pour
some hot water into the barrel, put your thumb on the barrel end (using a bit of
cloth, as the barrel may become hot) and shake the barrel to dissolve black
powder remains on the inside.
Ram down a dry piece of cloth (linen) to clean and dry the inside of the barrel.
Don't fix it to the ramrod, as then they may get stuck.
Use your worm or ball puller (tire-balle) to get the bit of rag out of the barrel, turning it slightly in the direction of the ends of the worm. This is a bit tricky, and may take some time. (Several re-enactors use a metal loop that they screw onto the ramrod to put the piece of cloth in, to avoid having to do it the hard way with the worm, but it is very unclear whether or not this is actually authentic).
Now it's time to tackle the rust on the outside of the barrel and the other metal parts. For this use a new bit of rag, moistened with olive oil. This rag is then repeatedly dipped into brick dust, and used as sanding paper. It might take a while (a long while....) to get this over and done with.
Grease or oil all metal parts with a bit of cloth or linen, greased with olive oil or preferably the bit of lard (or fat) that you have been issued with.