Cleaning a grease/cosmo packed rifle: (might be some newbie facts here so bear with me)
You'll need to disassemble the rifle down to the receiver. Parts that need an initial cleaning are the receiver cover, the entire recoil spring assembly (you'll have to take it down to the spring, c-clip, spring guide) bolt carrier, bolt (you'll have to take it down to the firing pin, extractor, and bolt housing) handguard, piston, operating rod, and operating rod spring. You need a degreaser at this point, and Simple Green which is safe for your hands and body will work just fine. If you want something more harsh you can use any kind of solvent, brake cleaner, etc., but Simple Green will do the trick. You can also use a product called Ed's Red. You make it yourself (recipe is here just search for it) For the barrel, you can use Hoppes 9 or Ed' Red. Once everything is degreased, clean and dry, you can re-assemble. Almost all the parts DO NOT require oiling or lubing except a few key places unless you plan on storing the rifle for months/year. I would suggest a light coat of oil on the sides of the bolt carrier that rub on the receiver and around the bolt however NO LUBE on the bolt face. You can even use lithium grease like you use on an M14. To prevent rusting of the barrel, you can run a patch down the barrel soaked in gun oil. When you're ready to go shooting, simply run a dry patch down the barrel to remove it.
For cleaning after shooting CORROSIVE AMMO (which is what the majority of us shoot in their rifle) you "should" strip the rifle down again to the level I described before. Cleaning the rifle now for use later prevents malfunctions (like cooking off a round/slam firing and jams) corrision, rust, and more accurate shots.
One way to clean all your parts is to first spray them down with Windex and scrub them. You can go to the Dollar Store and get $1.00 Fakedex and it will work just as well. You're only job here is to neutralize any salt deposits on the metal parts. You can do the same for the barrel. The barrel will have copper, powder, carbon, and mercury fulmite deposits on it but at this point you're only interested in stopping corrosion. The barrel will require bore cleaner (Hoppes 9 or Ed's Red) and multiple steps to clean it as deposits in the barrel are usually layered. After cleaning the parts with Windex, use the Simple Green and give everything a good scrubbing, receiver included.
Another way is to get a automotive plastic pan, fill it with Ed's Red or Hoppes 9 (which is a lot of Hoppes 9 this is where Ed's Red is cheap and excellent to use because it's cheap to make) and drop all your parts in there. This is a better process because the solvent gets into all the nooks and crannies of each part. While they are in there, you go after the receiver. Spray it down with Windex then degrease/clean it with Simple Green. For the barrel, follow your usual steps to get rid of all the deposited layers using Hoppes 9 or Ed's Red. Once you're done, wipe dry all the parts that were in the plastic pan, re-assemble, and lube. You can re-use the Ed's Red/Hoppes 9 until it turns a dark color. (meaning it's saturated with dirt)
Can a dirty SKS malfuction? ABSOLUTELY! Jams and slamfires are more prevalent with a dirty rifle. I have personally experienced both.
When you're done your SKS is clean, will shoot better, be more accurate, malfunction less, and even have a nice minty smell

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I know this process is a pain in the ass. As an SKS collector I don't shoot "all" my SKS's because of the process in cleaning them to maintain them properly when shooting corrosive ammo. I have a few setups I shoot all the time, and they satisfy my needs to shoot different SKS's (Chinese, Tach Driver, Russian).
Hope this helps.