A camera and time dilation? If I travelled near a black hole, my time would progress slower relative to someone on Earth. This is clear enough. However, what if we sent a probe with a camera to a black hole? When we watch the screen, would we see time through the camera's perspective — that is, would the Universe appear to progress faster as the probe got closer and closer to the black hole?
If I travelled near a black hole, my time would progress slower relative to someone on Earth. This is clear enough. 


Yes, no problem with the gravitational time dilation. 


  However, what if we sent a probe with a camera to a black hole? When we watch the screen, would we see time through the camera's perspective — that is, would the Universe appear to progress faster as the probe got closer and closer to the black hole?


No. We'd see the universe progressing normally, because we aren't subject to that gravitational time dilation. (I presume we're at a safe distance). Let's suppose it was a TV camera that took a picture 25 times a second and sent it back to us, adequately catering for redshift. The camera starts off taking 25 pictures a second as measured by us. But after a while we notice we're only getting 24 pictures a second, then 23, and so on. We see events in the wider universe progressing at their normal rate, but eventually the movie starts getting jerky as the frame-rate reduces. In the end the frame-rate reduces to zero, and that's the end of the show.