This $15 screen-cleaning solution is used in Apple stores, and I finally understand why
Whoosh 3.4 fl oz screen cleaner with microfiber cloth removes oils and smudges effectively without alcohol or ammonia, using a double-sided microfiber cloth and milky, non-toxic formula.
Wipe That Sneeze: How to Clean a TV Screen or Computer Monitor
Clean modern screens with a soft, dry microfiber cloth; use distilled water or mild soapy water for stubborn grime, and avoid solvents and excess moisture.
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Our screens are on a second-floor balcony. There really isn't any good way to clean the outside of the screens. Luckily, the fibers on this screen cleaner are small enough that they seem to fit through the screen and clean both sides.