Soft damp cloth with a drop of Dawn. The Dawn provides lubrication while breaking down oils and organics on your screen as you scrub. Give it a good scrub, should barely make any soapy bubbles. Wipe clean/dry with another cloth. I do about the same with my cheap reading glasses, a drop of Dawn, rubbed in with my finger then transfer to both lenses both sides, use the fingerprint of my fingers to scrub the lenses under a small stream of running tap water. When you can almost feel them squeak across the plastic lens you know they are clean. Wipe dry with a clean cloth.
My kindle screen looks like new (never used a protector) and is 6 years old, used heavily, daily. Cheap plastic lens readers look like new, no microscratches at all, last until the cheap hinges break or I accidentally step on them or other dumb move.
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