If the front of a spaceship crosses the horizon of a black hole in a matter of seconds, what does the back of the spaceship see that is still outside?

Lets assume that we are not talking about a real spaceship, because in the real world, the tidal fields of most black holes would shred the ship apart long before any interesting relativistic effects happened.

The time dilation in question occurs for the distant observer viewing the ship, not for the person in the rest frame of the ship assuming for the moment that the ship is smaller than the scale of the tidal field. The closer you are to the event horizon, the more time dilation will be seen by the distant observer, but also the more redshifted the light will be. So, the clock face and apparent motion of the front of the ship, as viewed from a distance, will appear to slow down, and the light from the ship shifted into the infrared. The back of the ship will be seen to move more rapidly and time progressing more quickly than the front of the ship. This emphasizes the well known problem that there are no rigid bodies in relativity.

In the mathematical limit, as the ship crosses the horizon, the distant observer will never observe this instant because the time dilation in the distant frame will be infinite, and the light from the front of the ship will be shifted from optical to infinitely long-wavelength radio waves. Also, there is a definite moment in the distant frame when they receive the last photon emitted by the front of the ship before it crossed the horizon. Meanwhile, the back of the ship will be seen still in motion at slower and slower speed.

You must note that to get the real large Doppler factors and time dilation factors, you have to be within MILLIMETERS of the event horizon. At greater distances, the relativistic effects are only moderate.