Billyy asked the Laptops forum for screen-cleaning advice. My instructions also apply to desktop monitors, HDTVs, cell phones, and tablets.
You don't need an expensive kit to clean an LCD screen. You can remove those smudges with common household items. And that goes for LED screens, as well, since they're simply LCD screens with a different type of backlighting.
You will need a microfiber cloth. If you wear glasses, your optometrist probably gave you one--although you might want a bigger one for a large screen. You can buy these for a few dollars from several Web sites, as well as from a photography store.
Use the dry cloth to gently wipe away the grime. If you encounter some stubborn dirt, apply gentle pressure; but not too much.
If that doesn't work, use a moist cloth. But how do you moisten it, and with what? You should definitely not use soap, a window cleaner like Windex, or anything containing alcohol.
Instead, get yourself an empty spray bottle (like the cloth, they're easy and cheap to buy), and fill it with a mixture of distilled water and white vinegar, in approximately equal parts.
Before you apply the formula, separate the screen from its power source. If it's a monitor or TV, unplug it. If it's a laptop, phone, or tablet, shut it down, unplug it, and, if possible, remove the battery.
Spray the water/vinegar mix onto the cloth, not the screen. Then wipe the screen gently, as I described above.
Wait until the screen is dry before plugging it in, reinserting the batteries, and powering it up.
You'll find additional suggestions on the original forum discussion.