Let $X$ be an arbitrary scheme.
I think one calls $X$ a characteristic zero scheme if all the residue fields of $X$ are of characteristic zero, for any point $x$ on $X$.
If $X$ is a scheme over $\mathbb Q$, then of course it is a scheme of characteristic zero.
My question is: does also the converse hold? And how does one make this rigorous?
Yes, you only have to show that any non-zero integer $n$ is invertible in $O_X(X)$ (then the universal property of localization induces a homomorphism from $\mathbb Q \to O_X(X)$ hence to the sheaf $O_X$). For all $x\in X$, as $n$ is no zero in $k(x)$, it is invertible in $O_{X,x}$, hence invertible in an open neighborhood of $x$. The local inverses glue into a global inverse by the uniqueness of the inverse.