If $G$ is a(ny) group, if $k$ is a field of characteristic 0, and if $V$ and $W$ are semisimple finite dimensional $kG$ modules, then $V \otimes_k W$ is indeed semisimple as a $kG$-module. This is due to Chevalley, and (I think I'm not off-base in saying this) inspired the characteristic $p>0$ result of Serre mentioned in other answers/comments.

The argument goes as follows: it is enough to prove the result after replacing $k$ by an algebraic closure. Now replace $G$ by the Zariski closure of its image in $GL(V) \times GL(W)$ -- this Zariski closure leaves invariant the same subspaces of $V \otimes_k W$ as does $G$, so we may suppose $G$ to be a linear algebraic group over $k$.

Since representations of finite groups in char. 0 are semisimple, a $G$-representation
is semisimple just in case that is true upon restriction to the connected component $G^0$. Thus we may and will suppose $G$ to be connected.

Finally, note that $G$ has a faithful semisimple representation, namely $V \oplus W$. Thus
the unipotent radical of $G$ is trivial so that $G$ is a connected and reductive group over $k$. Now the semisimplicity of $V \otimes W$ follows (*every* finite dimensional rational representation of $G$ is semisimple).

`$G$`

is (connected) reductive; moreover, such groups often have faithful irreducible representations but only tori are linearly reductive. On the other hand, in characteristic 0 linearly reductive = reductive. (And I guess the modules here are all finite dimensional?) Note too that the additive group has a faithful irreducible 1-dimensional module. – Jim Humphreys Mar 9 '11 at 21:37