If you can, break it down to the point where you can take off the furniture.
How to take apart an AK-47
Get yourself a couple old pans to keep the metal parts in, a couple old toothbrushes, some paint brushes, and a can of Mineral Spirits (from the hardware store). You can even use the Mineral Spirits on the wood, just don't let is soak in it for too long or you'll strip the finish. Use a couple of those green scrubbers (think pots and pans scrubbers), dip those in the Mineral Spirits and just lightly scrub. I keep a couple rags handy to wipe off the mess and see how things are going. You'll find that the Mineral Spirits will disolve the Cosmoline pretty quick, and the goop will be flowing, so change the Mineral Spirits when it starts getting crappy.
If you don't feel safe using the Mineral Spirits, I've also used WD-40, even Simple Green and lots of HOT water works well. Just remember how you took everything apart (pretty simple to re-assemble an AK - lots of videos on You-tube, too). And when you reassemble, take your time, make sure to wipe everything down with some CLP as you reassemble, lube and oil where it needs it. After the furniture air dries, you can also clean it up some more with some fine steel wool, followed by a tack cloth to get all the dust off. Then you can re-finish it the way you want, either with satin-finish polyurethane or just Mineral Oil to keep the stock look.