Don't disturb the asbestos if you can avoid it.
Check with your town to see if there are special rules, but in general the process will be: clean the existing tiles thoroughly, with something that will remove grease and dirt. If any old tiles are loose, nail them in place with 6D nails, then use a nail set to countersink the nail heads. Use a thin coat of floor leveler on the old tile to fill the seams between tiles and level off any dents and depressions that have formed over time. When it cures, lay down plywood as a new subfloor.
Check the directions for the new tile to see what they recommend for thickness and type of subfloor, as well as the recommended gap between sheets of plywood, and the number and type of fasteners to use to attach the subfloor. Be sure your fasteners are long enough to get all the way into the original subfloor, under the asbestos floor (unless you're really fond of squeaky floors and popped tiles).
Sweep the subfloor, then use a thin coat of floor leveler to fill the gaps between the plywood sheets as well as the dimples created by the fasteners. When it cures, clean the subfloor thoroughly, then lay your new floor. If you leveled things well, and fastened the new subfloor properly, your new floor should be fine.
I think most people picture ceramic tiles with curved edges when you say "tiled floor" rather than flat linoleum tile, which is what I'm guessing is actually there. Trying to put a new floor over ceramic tile would be a recipe for failure, but putting it over linoleum would be fine.