Is the screen they’re looking at a PsychoPy window or do they go into a different tab? You can use software like OBS to record the entire screen, including PsychoPy, or you can get just the PsychoPy window using win.getMovieFrames each frame and then win.saveMovieFrames to save the frames.
Overlaying the eyetracking trace is something you’ll need to do after as I presume you don’t want the participant to see it, you could do this in a separate experiment which just has a 
Movie component showing the saved frames and a 
Polygon whose position is the recorded eye position at that time.