Note that, since $G$ is a finite group, there is an invariant inner product on $V$. The results we need can be derived from the Schur orthogonality relations.
In the irreducible case, $V$ has an invariant complex structure iff the character function has $L_2$-norm equal to $\sqrt2$ or $2$ (otherwise it is equal to $1$). In fact, $V\otimes \mathbb C=U\oplus \bar U$ for a complex representation $U$ and $\chi_V=\chi_{V\otimes\mathbb C}=\chi_U+\chi_{\bar U}$ where $\chi_U$ and $\chi_{\bar U}$ are orthogonal (unit) vectors in case $U$ and $\bar U$ are inequivalent, or $U$ and $\bar U$ are equivalent and then $W$$V$ admits even a quaternionic structure.
In the general case, the criterion is that the irreducible components that are not as above must occur in (equivalent) pairs $(W,W)$, as Qiaochu wrote. On $W\oplus W$ we have $\left(\begin{array}{cc}0&-\mathrm{id}\\\mathrm{id}&0\end{array}\right)$ as invariant complex structure.