Is it true that if $\phi_1$ and $\phi_2$ are characters then $\phi_1\phi_2$ is a character of a representation?
I think this is true, say for instance if $R_1$ is a matrix representation with character $\phi_1$ of $G$ and $R_2$ is a matrix representation of $H$ with character $\phi_2$, then we can say $\phi_1\phi_2$ as the character of the matrix representation of $G\times H$ if we define $\phi_1\phi_2(g, h)=\phi_1(g)\phi_2(h)$. Will this suffice to prove our claim?
Suppose $G$ acts on a vector space $V$ with character $\phi_1$, and $H$ acts on a vector space $W$ with character $\phi_2$. Then $G \times H$ acts on the vector space $V \otimes W$ with character $\Phi(g,h) = \phi_1(g)\phi_2(h)$. Moreover every irreducible complex representation of $G \times H$ is a tensor product of an irreducible representation of $G$ with one of $H$. This is what is often called the external tensor product of representations. It is external since it takes in representations of two groups and spits out a representation of a third.
There is also an internal tensor product of representations which takes in two representations of $G$ and spits out a third. It is related to the external tensor product as follows: We take two representations of $G$ and get a representation of $G \times G$ via the external tensor product, then we restrict this representation to the diagonal copy of $G$ in $G \times G$. Understanding how these internal tensor products decompose is a much subtler problem, which is referred to in general as the Clebsch-Gordan problem.