The short answer is "no", but to be honest the question deserves a bit more of a reply than that.
For example, it's worth discussing what it would mean to "see" anything, which usually means registering light or other EM radiation. In that case, it's clearly impossible, because the very nature of light prohibits it from travelling faster than itself. On the other hand, maybe you mean "seeing" by using some hypothetical new particles, which invites a discussion about tachyons.
Tachyons are hypothetical particles that travel faster than light, but what is weird about them is that they *always* travel faster than the speed of light, and the effect of this is to allow them to move, effectively, backwards in time. Maybe tachyons don't actually exist, but assuming they did it would be difficult to see how you could make use of them -- in effect, tachyons are seen before the event that causes them, which violates all sorts of current laws of physics.
The point isn't to be discouraged by these answers, though, but to see them as a jumping-off point for exploring even more interesting questions. In all likelihood we will never see the inside of a black hole, or at least not if we wanted to live and tell anybody else about it, but it's worth pondering all the same, and hopefully you can enjoy reading up about, among other things, light cones, causality, spacetime, tachyons, event horizons, the gloriously-named "Cosmic Censorship Hypothesis", and so on.