To clean acrylic and plexiglass that has slight blemishes or dirt, start off by clearing away any debris.
Then, use a premium micro-fiber cloth with soapy water to clean the entire surface area. After wetting the cloth, be sure to lightly blot the surface, rather than applying pressure as you wipe. Too much pressure can actually result in more scratches.
After you have wiped the entire piece of acrylic, use the dry side of the cloth to dry or buff the area. Buffing or polishing the acrylic surface reduces future stains. If there are any remaining streaks, rewet the cloth and dry again.
Choose Carefully: There are a couple of cleaning items that you'll want to stay away from. In fact, using the wrong cleaning product can completely damage your acrylic displays to the point that they become unusable!
Do not use paper towels because they will not remove scratches and may even cause them. Stick with a micro-fiber type of cloth.
Avoid ammonia-based products, like Windex or other home glass cleaners, because they contain harmful chemicals that will actually damage the surface leaving it cloudy looking. Soapy water is the safest and most effective cleaning solution for acrylic.
You can use Acrylic or plastic cleaning solutions such as Novus; these will clean and also polish.