If it's a touch up, I just fog up the screen with a breath and give it a rub with a cotton t-shirt or cloth.
If I Need to clean a whole monitor I use a product called Novus 1. I don't know if it's available to the public though cause I snatched this one from Sony.
Simple green is non-ammonia based product I use to clean everything from degreasing engines, to cleaning windows. It's non-toxic biodegradable cleaner found at automotive or even grocery stores. U may wish to dilute it with distilled water because it's usually concentrated.
Just make sure you use a soft cottton terry cloth or a micro fibre rag so you don't scratch any AR coating that may be on the screen. An optometrist would have cleaners that are safe to AR/static coatings designed for eyeglasses that should work well.