Any chrome surfaces actually consist of an extremely thin layer of chrome plated onto another material. Regardless of what that other material is, from plastic to steel to aluminum, this chrome plated is what needs to be addressed when the time comes to clean it. You can always buy a special chrome cleaner, but there are several items in your home that work just as well and are far less expensive.
Spread a thin layer of toothpaste on the surface of the chrome plating. Use the opaque white toothpaste, not one of the gel varieties.
Rub the toothpaste-covered chrome plating with a soft, clean cloth in small circular and swirling patterns. There is no need to apply a lot of force.
Take a fresh cloth and wipe away the toothpaste, revealing a shining, clean surface.
Pour undiluted distilled white vinegar into a spray bottle.
Spray the surface of the chrome plating with the undiluted vinegar.
Wipe down the surface with a clean cloth.
Baby Oil Method
Dampen a portion of a soft cloth with baby oil.
Rub the baby oil into the chrome plating in the same manner as you did with the toothpaste.
Wipe off excess baby oil with a fresh cloth or a still fresh portion of the same cloth.
Wipe down the newly cleaned area with soapy water to remove any greasy feeling from the baby oil, if desired.