There are a few Browning BAR rifles out there. What kind is yours, Belgian, Pre 91, Post 91, LongTrac, ShortTrac? See what I mean? Each one functions similarly, but there are little tips that are specific for each one.
If you are able to take any rifle completely apart, cleaning is made easier.
Browning rifles come apart relatively easy, but I am saying this after 20+ years of doing it.
I think the biggest thing to concentrate on is the chamber and barrel. If you have to go in through the muzzle, pay strict attention on the crown. I use a good copper solvent with a good quality bore brush. Swap the brush for a cotton patch and clean until the patch comes out relatively clean. Make sure your chamber is clean and lightly oiled.