Yes you should clean: soap and water or any other mild cleaner is good.
You may not need additional primer, but you should fix any dents/scratches -- they will stick out after you put the new paint on. Auto-body filler (eg Bondo) works for dents and deep scratches. Before you prime, the whole door should be lightly sanded using 120-150 grit sandpaper to roughen the surface, and then wipe the whole door down to get rid of any particles from sanding.
Use an oil-base primer, and again, be sure to lightly sand first.
You might need to use more than soap and water - but that would be a first step. However, you don't want anything too harsh. Just makes sure that the surface is completely dry before painting.
An additional coat of primer might be worth it - but only if there are any places where the original primer has rubbed off or you have knocked the door. Places where the metal has been exposed should be sanded to remove any rust.
If the primer is sound then you should only need to undercoat before applying the gloss to give the finish some "body". I normally end up with 2 coats of top coat to give an even finish.