How do you put tile onto an old dresser top that has been converted into a bathroom vanity?

You can use a vinyl adhesive and a trowel to add the adhesive to the dresser top.

you will need at least 4 pole clamps and you will need at least 2 flat wood strips to clamp on either side of the dresser- the strips must stick above the dresser top by as much as the thickness of your tiles-- if the tiles are 1/4 inch thick then the flat stick must be raised 1/4 inch above the sides of the dresser and clamped.

The poles and clamps are only to give you a straight-line edging

Trowel the adhesive onto the dresser top --- and trowel adhesive onto the tiles with 1/8 "V" notched trowel and wait for about 15 minutes before mating the two-- onece you mate the two pieces they will be practically imobile (un-movable). Remove the flat wood strips. After the tiles are on you can add sanded grout -- fill the seems of the tiles and sponge off the excess-- Let it cure for about 24 hours before using and you''re done.

You may still need a wet tile cutter or a Glass cutter for cutting your pieces and you may need tile "plier snips" and or Ceramic Breaker Pliers.

Before embarking on this journey I suggest you watch HOW TO VIDEOS on "YouTube" type "Ceramic tiling" into the Search Bar or google "ceramic tiling How To Videos"