This is easier to do in general than in the special case, so I'll just generalize. Most of the claims below require proof, by the way!
Suppose $M$ is a smooth manifold and that $G$ is a group which acts on $M$ smoothly and properly discontinuously. Let $N=M/G$ be the quotient topological space and $\pi:M\to N$ the quotient map. Since the action is properly discontinuous, $N$ is Hausdorff and the map $\pi$ is open —in particular, the second-countability of $M$ implies that of $N$.
Let us now construct an atlas.
Let $p\in M$. Proper-discontinuity implies there is an open neighborhood $U\subseteq M$ of $p$ such that $U\cap gU=\emptyset$ for all $g\in G\setminus\{1\}$. The restriction $\pi|_U:U\to N$ is an homeomorphism onto its image, which is open in $N$. By replacing $U$ by a smaller open neighborhood of $p$, we can suppose that there is a chart $\phi:V\to\mathbb R^n$ in the (maximal) atlas of $M$ with $U\subseteq V$. We define $\psi_p=\phi\circ(\pi|_U)^{-1}:\pi(U)\to\mathbb R^n$.
Now the set $\mathcal A=\{\psi_p:p\in M\}$ is an atlas on $N$.