Is there any better than (2/k)-approximation algorithm for Independent Set in Coloring graph?
If you have a graph with maximum degree $\Delta$, then the greedy algorithm finds a coloring with $\Delta+1$ colors, so for $k = \Delta+1$ the assumption that you are given a proper $k$-coloring does not change the complexity of the problem. It is known that independent set is NP-hard to approximate within factor $\Delta^{1 - o(1)}$, and Unique Games-hard to approximate within factor $\frac{\Delta}{O(\log^2\Delta)}$, which, for $k=\Delta$ is within a $\log \Delta$ factor from the upper bound you cite. (I prefer to write approximation factors as bigger than 1, so I am thinking of the upper bound you cite as $\frac{k}{2}$.) 

However, it's still interesting to close these logarithmic gaps. This paper by Bansal may contain useful ideas.