Don't have much to add to those above but be sure to thoroughly remove the old finish before starting with your stripper of choice-MEK works the best but is ghastly in its health warnings and you need to strictly follow them. Other chemical strippers may take more time. Make sure to remove the old finish from checkering as this can cause problems--depending on the stripping solvent--you will have to use a tool that works with that solvent--like MEK will disintegrate a ordinary toothbrush into your wood. Personally, I use SoyGel as it is based on soybeans and pretty non-toxic with no nasty odors--it also sticks to the surface but it does take time to remove polyurethane.
You can also remove the finish via sanding but that has its own issues. If there is polyurethane/varnish etc. left on the stock, it will cause blotching of the stain which will ruin the effect of refinishing.
If you have a wood crafter's shop like Rockler or Woodcrafters or Amazon, you can buy pen blanks (small pieces of woods used for turning writing pens) and they do have birch. You can mix staining materials until you get what you want--basically the staining material is floated in a evaporating solvent leaving the staining material behind on the stock. When mixing stains, I always test it and the pen blanks are useful for that and small stock repairs in general.