Remove the drawers from the dresser.
Reach inside and unscrew the fasteners to remove the tabletop shelf from the furniture body.
Rub chalk over the ends of the plumbing fixtures extending from the wall.
Push the dresser into place, allowing the chalk to mark the back as places for plumbing access.
Pull the dresser back out and cut holes in the chalk-marked areas using a drill fitted with a hole-cutting attachment that creates an opening slightly larger than the plumbing pipes.
Install a suitably sized countertop and basin, and attach a faucet. Connect the plumbing, or call a plumber to install the water lines.
Detach the face of the top drawer. Glue and screw the face permanently back in place without the drawer “box.” Alternatively, attach a tip-out (a narrow bin for storage of small items) to the backside of the drawer face, and install it back on the dresser with spring-loaded hinges.
Slide the lower drawers back into the cabinet to store towels, washcloths and other bathroom items.