If $X$ and $Y$ are $G$-sets and $X \times Y$ is a G-set by $g \cdot (x,y)=(g \cdot x , g \cdot y)$. \pi is the corresponding permutation representation. Prove that $\pi_{X \times Y} \simeq \pi_X \otimes \pi_Y$.
Now $\chi_\pi (g)$ is the number of fixed points of $g$ in $X$.
So LHS: $\chi_{\pi_{X \times Y}}(g)=$ $\{$ number of fixed points of $g$ in $X$ $\}$$\{$ number of fixed points of $g$ in $Y$ $\}$
and RHS: $\chi_{\pi_X \otimes \pi_Y}(g)=\chi_{\pi_X}(g) \cdot \chi_{\pi_Y}(g)$ =$\{$ number of fixed points of $g$ in $X$ $\}$$\{$ number of fixed points of $g$ in $Y$ $\}$
So Im guessing that we can extend this to $\chi_{\pi_X \otimes \pi_Y}=\chi_{\pi_{X \times Y}}$
but this does not determine uniquely that $\pi_{X \times Y} \simeq \pi_X \otimes \pi_Y$.
The problem is more straightforward: The action of $G$ on the vector space $k[X\times Y] = k[X]\otimes k[Y]$ (where $k$ is your ground field) is given by $g(x\otimes y) = gx \otimes gy$, which is precisely the definition of the tensor product of representations.