CRT monitors are dust magnets, literally. Electrical fields emanating from the monitor attract dust particles like a flower attracts bees. Fortunately, cleaning is easy. Unplug your monitor and clean the screen gently with a soft, clean cotton cloth moistened with water. Never spray anything directly on the monitor. To dry the screen, use another clean, dry cotton cloth.
Monitor manufacturers say you should never use any chemicals (glass cleaners included) on your screen. The most important thing is to avoid ammonia, a component of most cleaners. The water-only method works very well, but if you really want to get a cleaning solution, buy a specific monitor cleaner based on your manufacturer's recommendations. Be sure to check your monitor's documentation first; if your monitor has an anti-glare surface or other delicate coating, it may require special cleaning procedures.
To remove dust on the outside of the monitor case, use the soft brush attachment and your vacuum cleaner. (Make sure your monitor is unplugged). Pay particular attention to gently brushing and cleaning the cooling vents. Avoid blowing dust into the monitor's case. If you find dirt marks on the case that won't vacuum away, wipe it with a cloth very slightly moistened with plain water. If that doesn't get the mark off, try the same amount of moisture with a bit of liquid dish detergent added. Avoid using kitchen cleansers on the plastics used in your monitor case.