For simplicity, let's say that the black hole is isolated and non-rotating (and uncharged), so that the situation is described by the comparatively simple Schwarzschild spacetime. Let's also suppose that the camera free-falls radially into the black hole.
What is the camera looking at? Suppose it is looking at some stationary object that does something with a known frequency. Your question is basically how at what frequency it will be observed on the video feed emitted by the camera.
Without loss of generality, we can suppose that the camera is looking at us, and that we're shining a laser beam at it: the 'doing something at a known frequency' would be the oscillations in the electromagnetic wave of the laser beam. We can do this because time dilation affects every physical process, so we might as well pick one that is more convenient to think about.
At this point, it is straightforward why the camera feed will not show any time dilation: being equivalent to a reflected laser beam, the gravitational blueshift when going inward will be cancelled by the gravitational redshift going outward.
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