Eli Chaps has done a few up in birch, I think however aniline dye was recommended to me at one time when I was refinishing the 30TKs birch stock. Because birch has a grain that is unique in properties a sander/sealer coat is recommended, this seals the grain with a clear coat so the stain, if used, will come out more uniform in color and blend in nicer. Check the reference section under woodworking to see if there's anything there to help you. Mr fixit
Did a search for that thread, here are some excerpts.
Just an fyi, but oil based stains on that birch stock will not get very dark. You might try a water based aniline dye that will penetrate and give you some color. A friend of mine had a M1-A that had a birch stock and we used a water based aniline dye and it really looked great. We found a color that I think was called Winchester Red. Those are great little rifles. I had one that had seen quite a bit of use that I traded for a 336 SC. I would love to have one in the condition yours is in.
Fiebings Leather Dye works great.
Ditto what Swampman said, I've heard good things about Fiebings, I just couldn't remember the name when I posted.
As Mr. Fixit said, I've seen guys use pre-stain conditioner with good results but I've always shied away from it for the simple fact that it is actually sealing the surface of the wood and that is going to inhibit the finish from really sinking into the wood. It's a trade off thing and I don't know if it's that big of a deal but I'm willing to put up with the darker spots to know the finish is working into the grain.
I mix and match stain colors to get hues I like. I just use plain ole Min Wax and I've been happy. I think lighter colors help bring out the natural golden-reddish hues that birch seems to exhibit. But darker colors can be nice too.
Here's some examples of birch stocks I've done with different colors: