All the CD's I put into the car's Bose CD player get scratches on the playing surface. I have cleaned the player 2-3 times using a CD lens cleaner without success.

Seems to me that the CD player is prone to dust going very easily.

Any tips?

I would try those aerosole cleaning cans and put the straw into the opening and blast away.

Air
 

Good idea, get a can of compressed air, any local PC store has it. Blast some air in and around the opening of the player. That should clean it up nice. As far as the scratches in the CD's. Use a good Automotive Polish and polish the scratches out. They'll be as good as new!!