the existing flooring is not damaged, if it is fill in with cheap vinyl tile just to make it even. I would strongly recommend to clean the floor very well with bleach or something to cut the grease and grime. You want a good adhearson between the old and new.
Then, NEVER lay the tile down symetrical or congruent over the existing tile seams. Do a one half overlap and then go on. In otherwords, the new tile would split the old tile's intersection lines.