Basic question about defining a character.

Question

I am reading Character Theory of Finite Groups, by Martin Isaacs. If $G$ is a fixed finite group, and $M_i$ is an irreducible $\mathbb{C}[G]$-module, then the book defines $\mathfrak{X}_i$ by choosing a basis for $M_i$ and letting $\mathfrak{X}_i$ be the resolting representation of $\mathbb{C}[G]$. My question is, how dose choosing a basis for $M_i$, result in a representation of $\mathbb{C}[G]$? I know a representation of $\mathbb{C}[G]$, is a homomorphism from $\mathbb{C}[G]$ into Aut($\mathbb{C}$).