group theory - What is the weight system for these ${rm SU}(5)$ representations?

I need to work out the weight systems for the fundamental representation $\mathbf{5}$ and the conjugate representation $\overline{\mathbf{5}}$. I'm not clear what this means. The $\mathbf{5}$ representation is of course just the representation of $SU\left(5\right)$ by itself. After picking a Cartan subalgebra as the diagonal matrices with zero trace, we can of course see that the roots are $L_i-L_j$ where $L_i$ picks out the $i^{th}$ element on the diagonal, and the weights are simply $L_i$ in this case.

It is supposed to be the case that I can use the weight systems of representations to show for instance that $\mathbf{5}\otimes \mathbf{5}=\mathbf{10}\oplus \mathbf{15}$.