Does $C_0(X)$ determine the topology for a locally compact space $X$?

Yes. If $X$ is a locally compact Hausdorff space, then $C_0(X)$ is an Abelian C*-algebra. Moreover, the Gelfand spectrum of $C_0(X)$ is homeomorphic to $X$.

In another word, if $C_0(X)$ and $C_0(Y)$ are C*-isomorphic, then $X$ and $Y$ are homeomorphic.

(In fact, every Abelian C*-algebra is of the form $C_0(X)$. This is known as the Gelfand Theorem.)

Let $X^\ast$ be the one-point compactification of $X$, with compactifying point $\infty$, then if $C$ is closed in $X$ and $x \in X\setminus C$, note that $C^\ast:=C \cup \{\infty\}$ is compact in $X^\ast$ and as $X^\ast$ is normal we can find a continuous $f: X^\ast \to \mathbb{R}$ such that $f(x)=1$ and $f[C^\ast]=\{0\}$. And then $g=f\restriction X \in C_0(X)$ and $g(x) \notin \overline{g[C]}$ and so the set $C_0(X)$ separates points and closed sets. By well-known general topology facts means that it determines the topology on $X$ (the topology on $X$ is the unique smallest topology that makes all functions in $C_0(X)$ continuous) and what you state about nets is a consequence of that fact.