Use your vibrating sander and aluminum oxide paper to dull the backsplash surface, just enough to remove the shine. Don't sand the grout. Wipe off the dust.
Put masking tape on the wall and counter top around the edges of the backsplash, to keep the paint off the surrounding areas.
Brush primer onto the tiles in a thin, even coat. Keep the brushstrokes going all in one direction. Don't paint the grout.
Let the primer dry for eight to 10 hours.
Brush your paint onto the tiles, again putting it on in a thin, even coat, with the brushstrokes in one direction. Don't try to get complete coverage of the paint, but concentrate on keeping the paint layer thin.
Brush on a second coat of paint in the same way as the first. Let it dry. Repeat with more coats until the surface looks the way you want it. Let the last coat dry for 24 hours.
Brush a coat of polyurethane over the tiles, using the same thin, light coat as with the paint. Don't brush it on the grout.
Let the polyurethane dry for eight to 10 hours. Dull the surface of it by hand, gently, with extra-fine sandpaper, so the next coat will stick. Wipe off the dust.
Brush on the second layer of polyurethane. Let it dry, sand it lightly by hand, and brush on a third layer. Let the third layer dry for 24 hours.