Secure plastic dropcloths to the wall under the windowsill with painter's tape. Put painter's tape around the sides of the sill, where it meets the wall. Place another dropcloth on the floor under the window.
Apply the chemical stripper to the sill with a clean cloth, following the package directions. Let the stripper sit for the instructed amount of time, and then scrape the paint off with a putty knife. Rinse the sill with water, if instructed.
Sand the bare wood of the sill with 120-grit sandpaper, following the grain of the wood. Wipe up the dust with a tack cloth.
Stir the wood stain with a paint stirrer so the color doesn't settle. Paint the stain onto the sill with a paintbrush, following the wood grain, using even strokes that overlap.
Let the stain soak into the wood for 10 minutes, then wipe up the excess with a clean cloth. Repeat the application if you want a darker finish. Let it sit for 10 minutes, and then wipe up the excess. Let the stain dry for 24 hours.