How to define a language for an independent set problem of a graph?
A language $L$ describes decision problem $p$ means that $\forall x \in L $ the instance $x$ is a Yes/Accept instance of $p$. 

For the independent set problem, we have to define language $L$ such that every $x\in L$ be an independent set of graph $G=(V, E)$. 

So one possible way to define decision language for independent set problem is
$$L=\{I\,|\, I\subseteq V ,\,\forall u,v\in I \, (u,v)\notin E\}.$$

If you wish to define the language generally you can write it as
$$L=\{(G=(V,E),I)\,|\, I\subseteq V ,\,\forall u,v\in I \, (u,v)\notin E\}.$$