Two microfibre cloths, available from any supermarket.
Don't get the 'smooth' ones specially for glass, they can be a bit hard on a plastic screen, get the 'fluffy' ones that make your fingers itch, like picking up tiny velcro.
One cloth completely wetted in lukewarm water [nothing else], then squeezed as dry as you possibly can [trick - fold in 4, then roll & wring. You really want it as dry as possible].
The other cloth totally dry.
If the screen is warm, you may need the first cloth very slightly wetter - but never anywhere near wet enough to dribble if squeezed - you really don't want any water that could run down the screen. The cloths work better when almost dry anyway.
If the screen is cold & also dark, i.e. computer off, you'll also find it much easier to see when it's clean.
Wipe the damp one gently over the screen, multiple times, turning the cloth, pay attention to the edges & corners. The trick is gentle repetition, rather than 'force'.
Then, whilst the screen is still very slightly damp, wipe over again with the dry one, completely drying the surface.
Smears gone forever… well, until next time ;-)
Please don't be tempted to use any type of domestic cleaner or alcohol. You never know what type of screen coating any given screen has & some cleaners can irreparably damage the coating layer of some screens.