(Revised somewhat to try and prevent misunderstandings.)
First: I'd like to discourage you from trying to interpret superpositions in terms of Schrödinger's Cat, when what you're trying to understand involves coherent unitary operations performed on the system. Considering superpositions of states of a massive number of particles makes it difficult to meaningfully consider different bases of measurement, which (among other things) is necessary to describe phenomena such as entanglement. (The fact that it prevents meaningful discussion of coherent evolution is basically the phenomenon of decoherence.) If we were to try and describe teleportation in terms of cats, we'd have to describe it this way:
1. What is put into the boxes are not live cats and dead cats, nor even two independent indeterminate live/dead cats, but a pair of cats A and B whose states of living are entangled with one another in a single pure state $|\Psi\rangle$ which is a superposition of one live cat and one dead, but where there is no definite fact of whether A is the live one and B the dead one, or vise versa.
2. The measurement we perform is not just to open a box to see a cat is alive; the pair of entangled cats are nothing but a correlated system which we will use to transmit more information. In this case, we are interested in transmitting yet another cat C, whose state $|\varphi\rangle$ of life/death is also indeterminate, but independent of A and B. We perform a quantum measurement corresponding to disentangling A and C (although they weren't entangled to begin with), and then observe whether they each of A and C is alive or dead (which will cause a collapse of their wavefunctions). Depending on the outcomes of these measurements — A alive and C alive, or A alive and C dead, or etc. 28
— we then perform two operations:
3. an operation which will toggle the state of B's life or death; either killing it or resurrecting it miraculously, or — more generally, as B may not be definitely alive or dead — rotating the state of B's health through an angle of 180° about a cycle of life and death;
4. another operation which does, er, something which I can't describe easily in terms of the life and death status of cats. Perhaps it changes the colour of its coat, or something; after all, C's coat may be differently coloured than B's was to begin with.
This is rather unweildly, and as you might imagine, doesn't actually shed any light on the matter. But this isn't surprising — Remember, <PERSON>'s Cat was a thought experiment which was designed quite deliberately, by <PERSON>, to be absurd. If you hope to get any intuition from it, you're almost certainly going to fail, and if you seem to get a simple explanation for what's going on by pretending a quantum system is a cat, you've probably missed some essential details.
Quite seriously, teleportation does involve two elements which are very much quantum: the fact that an entangled resource state is used — two particles A and B, neither of which are the system whose state is being 'teleported', in a joint state $$|\Psi\rangle_{A,B} \;=\; \tfrac{1}{\sqrt 2}\Bigl( |0\rangle_A|1\rangle_B \;-\; |1\rangle_A|0\rangle_B \Bigr) \qquad\quad \Bigr(\text{or a similar state}\Bigr)$$ — and that the state $|\varphi\rangle$ of a third system C can be transmitted to B, by performing a <PERSON> measurement on A and C, which can be simulated by performing the transformation $(H \otimes I \otimes I )(\text{CNOT} \otimes I)$ on the state $|\varphi\rangle_C\otimes|\Psi\rangle_{A,B}$ and measuring $C$ and $A$ in the standard basis, and then performing simple single-qubit operations on B depending on the outcome.
The information about the measurements of A and C have to be transmitted to B somehow; these are sent as classical signals, and travel only at the speed of light. Until they arrive, the system at B is essentially random: whoever had the particle A knows what state it's in, but it may not be the original state of C, and whoever owns B has no useful way of doing anything with it until they know what operations they have to perform to recover the state of C. However, despite the fact that the measurement outcomes on A and C are 'classical', the state which can be recovered with them is fully quantum. 682