Ok... When drilling into melamine you want to drill holes before you insert screws. It sounds like you attempted to just drill the screws themselves through the melamine.

To keep the whole thing centered and spaced correctly most people just turn the whole thing upside down and build it upside down. If you put the top shelf on the bottom then upside down tub then cardboard then the next shelf and repeat. The sheets of cardboard will insure that you have a proper amount of space between the shelves to allow the tub to slide freely in and out. Drill the holes and screw the screws in as you stack