Why are there no stars on New Horizons images of Pluto
The answer by @mso is very good. Another contributing factor is the exposure time of the camera on New Horizons. 

The amount of light reflecting from Pluto is greater than that from the background stars. As a result the exposure time is set to achieve a good image for the desired target and detail required. 

Over the integration time the amount of light from background stars is not enough to register onto the focal plane and subsequent image. 

In order to get good pictures of stars one needs to integrate over long period of time while maintaining accurate pointing.