It's best to start with the most gentle method for cleaning the monitor and progress to other methods if necessary. Start by gently dusting the monitor, using a can of compressed air, and avoid contacting the screen if possible. If this is not sufficient, use a microfiber cloth to wipe the screen, while applying light pressure.
Before cleaning the monitor, turn it off and let it cool down. If using any liquids, unplug the screen as well. Cool screens are less likely to become damaged. Avoid using a circular buffing motion, and make broad strokes in one direction at a time. If a dry microfiber cloth doesn't clean the monitor sufficiently, try using a slightly dampened cloth to remove stubborn dust or fingerprints. Make sure the cloth is only slightly damp; if enough water collects to form a drop that can roll down the screen, it might travel under the bevel, where it can cause damage.
For even tougher marks on the screen, create a mixture of half distilled water and half vinegar. This solution can cut through difficult substances that collect on the screen. Avoid using alcohol- and ammonia-based solutions, as they can damage the screen and protective coatings.