Given two locally compact topological spaces $X$ and $Y$, and a local homeomorphism $f : X \to Y$, Gelfand duality gives us a homomorhism $Cf : C_0(Y) \to C_0(X)$. How does the fact that $f$ is a local homeomorphism manifest itself in the homorphism $Cf$ between $C^*$-algebras?

Your first claim seems to need a small correction. In general, the continuous maps $X\to Y$ that induce homomorphisms $C_0(Y)\to C_0(X)$ are those which are proper, i.e. where preimages of compact sets are compact. (Just take $X={\mathbb R}$ and $Y={\mathbb R}/{\mathbb Z}\cong {\mathbb T}$ to see what can go wrong otherwise.)

So your question is really restricted to local homeomorphisms from one locally compact space to another that are proper. These are then forced to be covering maps (see Wikipedia) and hence $f^*: C_0(Y)\to C_0(X)$ has to be injective, hence an isomorphism onto its range. Martin Brandenburg's answer here explains that the converse is true, namely that a C*-subalgebra of $C_0(X)$ has to correspond to some compact Hausdorff quotient of $X$.

EDIT: see also this earlier MSE question