Is there a good way to get the dander and hair that gathers...?

Question

on the back of the hairdryer, where the air gets sucked in?

Answer 1

Back when I had hair (!!) I used an old toothbrush to brush the lint and stuff out of the built-in screen on the back of the hair dryer. It worked well and was a simple solution for the particular design of the hair dryer I used.

Answer 2

I put a filter on mine. I used an air filter from something else and trimmed it to fit inside the backside of my hairdryer.  You could use one too.  If you can't put it inside without hindering the effects of the dryer use it on the outside with a rubber band.

Periodically I clean it by washing it out in the sink or replace it.