Sand the front, sides and drawer fronts of the dresser just enough to scuff the surface so paint will adhere to it. Paint the dresser using enamel paint that will stand up to the humidity of the bathroom. If you prefer to leave the dresser unpainted, apply several coats of polyurethane to protect it against humidity.
Set the sink upside down on top of the dresser and draw the outline in pencil. Drill a hole in the center of the outline, and, beginning at the hole, use a jigsaw to cut out the space for the sink. Cut along a line ½ inch inside the outline so the sink drops into place but has space for the lip of the sink to rest on.
Mark the back of the dresser at the spots the plumbing pipes come out of the wall. Use the drill and jigsaw method to cut holes for the pipes in the back of the dresser.
Remove the top drawer of the dresser if it interferes with the plumbing. Take the face of the drawer off and discard the drawer. Glue the face of the drawer to the dresser, or hinge it at both ends, and build a tray inside that opens when the drawer face opens.
Tile the top and top edges of the dresser. Apply a layer of mortar, set the tiles, and let the mortar cure for a day before adding grout and grout sealer.
Remove the old vanity from the bathroom and slide the dresser into place, making sure the plumbing protrudes into the inside of the dresser. Use a caulk gun to run a bead of silicone caulk around the cutout for the sink. Drop the sink into place, wipe up caulk that oozes out, and hook up the plumbing.