How to tile over vinyl
Old vinyl floors in bathrooms are a pain to tear up, and if the vinyl was installed before 1980 it may even contain asbestos. However, if you don't mind raising the height of the floor a little, you can usually tile right over it, even directly on the vinyl, and thus not have to deal with the asbestos issue at all.
Ceramic tile requires a stiff base to keep it and the grout from cracking. So the first thing you have to do is check the thickness of your floor. You can usually figure the thickness by pulling up a floor register or removing the door threshold. If the ceiling is open below the floor, you can often tell from where plumbing penetrates the floor. As a last resort, remove the toilet and examine the area around the ring; you'll have to pull the toilet anyway at some point.
If your floor framing is spaced 16 in. apart, the combination of subfloor plus underlayment (a second layer of plywood directly under the vinyl) should add up to at least 1-1/8 in. If it's 24 in. apart, it should add up to 1-1/2 in. If the floor is less than this, it probably isn't stiff enough to lay tile directly on vinyl. In that case use tiling Method 1, and add either 1/4- or 1/2-in. cement board to build it up. Keep in mind that in doing so, you'll be raising the floor level 1/2 to 3/4 in. (cement board plus 1/4-in. tile), which means that you'll have to trim the door, raise the vanity, extend the toilet ring, and make a new transition to the hallway.
If your floor is already stiff enough, you can lay the tile directly over the vinyl using Method 2. With this method you only build your floor up 1/4 in. However, if you choose this method, you should be aware of the asbestos issue. Asbestos is a known carcinogen that was used in many products including vinyl tile, asphalt tile, sheet flooring and adhesives made until 1980. So if your floor was laid after 1980, it won't contain asbestos unless the installer used older materials. You can clean and sand it to improve tile adhesion, or even tear it out.
However, if you have an older home, and don't know when the floor was laid, do not sand it or disturb it. Simply strip off the old grime and wax with an ammonia-based cleaner. When it's dry, apply a little tile adhesive and let it dry to test for good adhesion. If thin-set mortar with an acrylic additive doesn't stick well, try a mastic-type adhesive. Both are available at home centers and tile stores.
In any case, tighten any loose flooring by screwing down the entire surface with galvanized wood screws spaced every 6 in. Add more screws in obviously loose areas.