Unless I drastically misunderstand your question, of course the characters $\chi_i$ depend on the representation $\rho$. Try looking at the simplest nontrivial case: $G = \mathbb{G}_m$ acting on a one-dimensional vector space. In this case, there is exactly one $\chi_i$ and it is simply a character of $\mathbb{G}_m$, i.e., is of the form $x \mapsto x^n$ for a unique integer $n$. This integer $n$ is determined by (and determines) $\rho$.
Unless I drastically misunderstand your question, of course the characters $\chi_i$ depend on the representation $\rho$. Try looking at the simplest nontrivial case: $G = \mathbb{G}_m$ acting on a one-dimensional vector space. In this case, there is exactly one $\chi_i$ and it is simply a character of $\mathbb{G}_m$, i.e., is of the form $x \mapsto x^n$ for a unique integer $n$.