How do I prepare a bathroom floor for tile?

Use Vinyl Floor Stripper

Apply vinyl floor stripper to the floor. This chemical will disintegrate any dirt and waxy residue. Use an abrasive tool such as a scouring pad to scrub the floor. The scratches left by the scouring pad will create a surface that will better adhere to the cement based thin-set.

Joists

Drive screws that are 2 1/2-inches in length into the floor joists. This ensures that the underlying sub-floor are secured in place. Go to the floor under the bathroom and drill a hole into the ceiling. Start at a corner and start drilling 1/4-inch holes. If you hit an air pocket, keep moving an inch at a time until you hit something solid. That is a joist. The joist is a wooden seam that keeps the bathroom secure into that particular floor of the house.

Loose flooring

Look for any areas where the vinyl loosened from the floor. Cut out any loose spots and fill the area with cement based thin-set. You are ready to install the backer board.

References:

Family Handyman: Install a Ceramic Tile Floor in the Bathroom

Resources:

This Old House: Home Improvement and Remodeling