A slightly errant brush, cloth, or vacuum can cause permanent damage, and without even touching the boards, a zap of static electricity can destroy a processor. Check your warranty before taking any office equipment apart. Conventional wisdom is that you should only clean a screen with a damp cloth, as many screens have an anti-glare coating that you risk damaging if you use a harsh household cleaner. Even traditional window cleaners can damage some coatings and screens, especially on laptop computers. Always check your manual.
Screen wipes are also available which are also great for the keyboard and general cleaning.
Wipe your screen first, then the keyboard, mouse, and casing. Stop your computer from getting dirty in the first place by having them handy to wipe the grease off your fingers after eating junk food. If you are using just a damp cloth to wipe the screen, be sure to dry it off as well to prevent streaking. When cleaning the monitor casing use a soft cloth with your cleaning fluid. Antistatic sprays and other specialty cleaning chemicals are very common, and are formulated so they will not cause damage to even the most sensitive screens.