Yes it's possible - and it can indeed collect dirt inside which will ruin the picture, especially if the TV comes from a smoking household.
First, you need to remove the speaker frets. They will just pull off, but be careful not to bend them too much - start from the lower end, the next two pegs are in the middle and two more at the top. Then, at the edge of the glass, above and below the speakers, you should see four small clips that you need to remove. Pull them off to the side with a small flat screwdriver or something similar. Under them, you will finally find the screws that keep the contrast screen in place.
BE EXTRA CAREFUL when removing it so you don't drop it on the floor - optimally, have someone to help and keep it in place when you remove the screws. The fixings are glued to the actual glass and might even snap off so the glass will be loose sooner than you expect...
Then, just clean both the glass and the picture tube surface, and put everything back on. Dishwashing liquid is about the strongest detergent you should use.