This answer takes a more general viewpoint than Alexandre's. The generality is in response to the small number of assumptions on the spaces involved.

First, you should assume that $Y$ is locally path connected. Only on special occasions does anything good come from looking at coverings of non-locally path connected spaces.

Covering maps are highly structured, so promoting a local homeomorphism (even with the locally compact assumption) to a covering map requires strong conditions. Even if $f$ has the following (rather strong) lifting property, $f$ is not always a covering map: For each map $g:W\to Y$ from a locally path connected space $W$ such that $g_{\ast}(\pi_1(W,w))\subseteq f_{\ast}(\pi_1(X,x))$, there is a unique lift $\tilde{g}:W\to X$, $\tilde{g}(w)=x$ such that $f\tilde{g}=g$.

Here are (necessary) conditions you must have for $f:X\to Y$ to be a covering map.

1) $f$ has **unique path lifting** (equivalently, if $P(X,x)$ is the set of paths in $X$ starting at $x$, then the induced function $Pf:P(X,x)\to P(Y,p(y))$ is a bijection for each $x\in X$.

2) $f$ has unique lifting of homotopies of paths (this doesn't follow from 1. and you can formulate it in the same way as a bijection on path spaces)

Properties 1) & 2) still aren't enough to promote $f$ to be a covering map. You need to strengthen unique path lifting.

3) $f$ has **continuous** unique path lifting if $P(X,x)$ has the compact-open topology and the induced map $Pf:P(X,x)\to P(Y,p(y))$ is a *homeomorphism* for each $x\in X$.

A surjective local homeomorphism with 2) and 3) and $Y$ locally path connected now has the general lifting property listed above but this is still not enough. Such a map is a semicovering (shameless plug - but it's open-access so its ok right?): Semicoverings: a generalization of covering space theory, Homology, Homotopy and Appl. 14 (2012) pp.33-63 ). There are semicoverings of the Hawaiian earring - which is locally compact - that are not coverings (example 3.8 in that paper). On the other hand, a covering map always has properties 2) and 3).

This suggests that in the end you will need to assume $Y$ is semilocally simply connected.

If you have $Y$ locally path connected, semilocally simply connected and $f$ has properties 2) and 3), then $f$ is a covering map. With these new assumptions on $Y$, I suspect you can weaken 3) to 1).