Let $\phi$ be an algebraic group homomorphism from $G_{m}$ to $GL_{n}(\mathbb C)$,where $G_{m}= \mathbb C^{*} $. Then image of $\phi$ lies in the $D(n,\mathbb C) \cap GL_{n}(\mathbb C)$. Moreover each diagonal entries will be of the form $t^{m_{i}}$ where $1\leq i\leq n$ and $t\in G_{m}$.
Now if the image lies in the diagonal matrices then it is clear that each diagonal entries will be of the required form as each algebraic group homomorphism from $G_{m}$ to itself will be of the form $t^{m}$ where $t\in G_{m}$. To show the image lies in the diagonal matrices I have only information that it is an abelian closed subgroup of $GL_{n}(\mathbb C)$.
As Jyrki Lahtonen mentioned, problem is up to a choice of basis. Use that $G_m$ (any torus) is consisting of semisimple elements and semisimple elements goes to semisimple under $\phi$. Hence the image $\phi(G_m)$ is consisting of commuting semisimple elements. So we are done.