Proper preparation cannot be emphasized enough - it is usually the key to a lasting and well laid floor.
It is absolutely imperative that the old floor be smooth and clean. Older vinyl must be removed completely as even very small pieces left stuck to the subfloor will show up in the new flooring. Any protruding nails, screws or small chips of wood that stick up must be removed. Glue from prior floors must be removed or sanded down. Long handled scrapers are available for the task of removing old vinyl flooring, and are not expensive.
Concrete floors must be treated with the same care. Protruding bits of concrete from poor finishing must be chipped off and small holes filled with grout. If the concrete was "broomed", with small grooves in the surface (a converted deck perhaps) it would be wiser to use the type of tile requiring glue as the glue will take up very small indentations in the subfloor.
Once the heavy work of smoothing the floor is finished, the surface needs prepared. A new covering of " plywood is ideal, though seldom practical. Kitchens often prohibit adding height to the floor as appliances such as dishwashers may no longer be able to fit under counter tops. If possible, though, and within the budget, additional plywood is recommended.
Shoe molding or baseboards should be carefully removed and saved for later re-installation. The new tile needs to go under at least the shoe molding; it is nearly impossible to cut the tile exactly up to such molding as few walls are absolutely straight, and any gaps will be easily visible. Better to remove and re-install such molding, even to the point of buying new shoe molding if none is present. It is not expensive and can add considerably to the appearance.
Clean the floor thoroughly. Sweep, then sweep again. Mop concrete floors well, and a slightly damp mop on wood floors is a good idea. Let dry completely and vacuum thoroughly. The goal is to remove absolutely all dirt and dust from the surface.
The next step is to seal the subfloor. Whether concrete or wood, paint on a layer of sealer appropriate to the surface, either wood or concrete sealer. Pick a time for this so that the floor will not be used from the point of applying sealer to laying the final tile if at all possible. Shut off any kitchen usage, for instance, and eat out for a day or two. If the floor must be used, make absolutely sure that no dust or dirt is tracked onto it - leave shoes in the next room. Any dust or dirt will affect the adhesive properties of the tile and can often lead to early failure of the self adhesive vinyl tile, letting it move or completely peel off. Keep the subfloor clean at all times.