The Setting: a shaded place, in cooler temperatures. Preferably covered parking or in the garage. If outside, early in the morning. Sitting in the car with doors and windows closed.
The Tools: I seem to have my best results with a microfiber waffle towel. A "regular" microfiber towel will work to. I've used an interior detailer which is safe for screens and a glass cleaner which is safe for screens. Each has produced fine results.
The Process: One or two sprays across the face of the microfiber. If pressing against the skin, it should feel moist, but not soaked. With light pressure, I move left-to-right, edge-to-edge. Do that as many times as needed. No circular motions. If there's a slight residue, flip the towel over to a dry side and do a pass to remove that.
The Checks: If you think the screen is clean, if in low light, hold a torch in the area of the screen and take a look at it. It should be clean with no streaks. Don't chase residual particles. There's always some tiny bit of something stuck somewhere.
Afterwards: The microfibers I use for screens are dedicated only for screens. This reduces the risk of contamination, catching something and then dragging it across the screen. You can have a stack of towels. When it's time to wash the towels, wash them by themselves, with a microfiber wash cleaner.