Short answer: No.
Short answer as a complete sentence: We have no reason to believe that there is any way that teleportation (or any other quantum mechanical effect, or any other physical phenomenon which we believe occurs) allows you to send superluminal signals.
Quantum teleportation can only be used to transfer quantum states. Those quantum states may encode classical information, but they are still quantum states. (Of course, the world being quantum mechanical, even "classical information" is represented by quantum states, albeit those of a very large number of particles at once — but never mind.)
Teleportation requires open classical communication to work anyway; so teleporting the information classically won't save you any work, and in fact is totally unhelpful — except as a quantum mechanical version of a Vernam cipher (i.e. a one-time pad).
Whether anything is secretly happening in quantum mechanics faster than the speed of light is actually a matter of philosophical debate in the foundations of physics. There are people who say that there is (such as advocates of de Broglie–Bohm theory), and people who say that there isn't (advocates of the Many Worlds Interpretation, Consistent Histories, and some Bayesians). What people mostly agree on is that quantum mechanics allows you to realise correlations in probability distributions which are not possible in slower-than-light local hidden variable theories; but that even if there is anything happening faster than the speed of light, you'll not going to be able to use it to transmit signals, because everything looks like correlated but uncontrollable random outcomes.
Related question: Why can't quantum teleportation be used to transport information?