Suppose $G$ is any finite cyclic group--generated by $g$, say--and $H$ is any other group. If $\varphi:G\to H$ is a homomorphism, then the order of $\varphi(g)$ must divide the order of $g$.
Now, given some appropriate $h\in H$ (in the finite $|G|$ case, such that the order of $h$ divides the order of $g$), we can define a function $\varphi_h:G\to H$ by $\varphi_h(g^k)=h^k$. This will be a homomorphism.
In fact, all such homomorphisms $\varphi:G\to H$ must have this form, for one can readily show by induction that $\varphi(g^k)=\varphi(g)^k$ for all integers $k$. Thus, the situation is completely described.
Note that in the particular case you've described, all $h\in H$ are appropriate, so fixing $g$ as one of the two possible generators of $C_6$, we may describe $6$ homomorphisms $C_6\to S_3$ as described above.