What was the original finish on the stock? What is the nature of the grime? That controls the choices of cleaning solutions.

For a military stock with just an oil finish, laquer thinner can be used, with firm rubbing. Once the grime and old oxidized oil is removed, fresh linseed oil can be rubbed in. Larry Potterfield from Midway Usa has a short video that describes that process.  But you wouldn't want to do that on a rifle that was varnished originally, unless the finish was pretty much already ruined.  (You need to Login or Register to view media files and links)

Here is a picture of the stock

My method has been to run it through a dishwasher set as hot as it will go, and using TSP (trisodium phosphate) as the detergent.  This will get ALL the old grime out, but you will have to re-oil the wood.  It'll be worth it, believe me!   

This is an update of the method of boiling the stock in a solution of Oakite that I found in an old gunsmithing book.

When raising dents in an old stock by heating with a hot iron over a damp wash cloth, often the heavy-use areas- wrist and fore end- will show up almost black and very ugly. Try soaking with Go-Jo hand cleaner. Reapply several time. Two-piece stocks will ugly up too sometimes. If it's really noticeable you might have to restain. Storing long guns with wood stocks muzzle-down was done years ago to lessen weak wrists from oil saturation.