Here's a simple and far-from-optimal condition guaranteeing uniform convergence.
Suppose for that there exists $C>0$ such that for any positive integer $n$ $\newcommand{\bR}{{\mathbb{R}}}$
$$ \int_\bR |f^{(n)}(x)| e^{-x^2/2} dx\leq C^n. $$
In this case the associated Hermite series is
$$ \sum_{n\geq 0} \frac{c_n}{n!} H_n(x), $$
where
$$ c_n=\frac{1}{\sqrt{2\pi}}\int_\bR f^{(n)}(x) e^{-x^2/2} dx. $$
and this converges to $f$ uniformly on compacts. This follows from known asymptotic estimates for Hermite polynomials.
For more precise results, you need to look at Gaussian-Sobolev spaces and the Ornstein-Uhlenbeck operator $H$.
