Why is there dirt/dust in my screen?

Use a plastic prybar to lift up the edges of the screen. You can use a little heat if needed, but it's not necessary. The plastic lens will come off. Then carefully clean the screen and remove the old adhesive. You can use a little alcohol to help you remove some of the old adhesive, and use a plastic pry bar to scrape some of the adhesive away. Be careful not to damage the LCD or PCB board. If you think you might damage them, then you'll want to remove them from the frame. That's more work for you, though, and it's not necessary, so avoid that if possible.

Then you can replace the screen with a new screen and adhesive.