Test the stain on three scrap pieces of birch plywood to get an idea of how dark the stain is and how many coats you need. Brush one coat on the first piece, two coats on the second and three coats on the third. Allow the stain to dry between coats and compare the three pieces after all are dry to identify which depth of stain your prefer.
Work in an area that is well ventilated. Brush the surface of the birch plywood with a dry, stiff bristle brush to remove any dirt and debris.
Sand the plywood with 220-grit sandpaper. Wipe the sanding residue from the plywood with a tack cloth.
Brush a thin coat of pre-stain wood conditioner on the plywood. Let it sit, then remove it following the manufacturer's instructions. This stain controller helps the wood absorb stain evenly and keeps the plywood from absorbing stain to the point the wood grain does not show on surface.
Apply a coat of stain to the plywood, following the manufacturer's package instructions. Use brush strokes in the direction of the wood grain, wiping the immediate area with a dry rag after every three strokes. Pull the rag across the plywood surface in the direction of the wood grain, keeping the look even and smooth.
Repeat the process in Step 5 until you have covered the plywood surface. Use as many coats as you determined necessary in Step 1 to achieve the best dark finish. Allow the stain to dry completely between coats.
Brush a coat of sealer over the dry stain. The staff at the home improvement center can advise you as to a compatible sealer for the type of stain used.