Hardness of finding a graph coloring given the optimal number of colors
Knowing the exact value of the chromatic number $\chi$ cannot help by more than a factor of $n$.  Since there are only $n$ possible values of $\chi$, you can 'guess' its value, i.e., run processes $P_1,\dots,P_n$, where $P_i$ runs an algorithm assuming $\chi=i$.  This whole scheme can find an optimum colouring in time at most $n$ times the time that it takes $P_\chi$ to find an optimum colouring.

On the other hand, if you're talking about parameterizing the running time by $\chi$, then it's a much more interesting question.  It's in FPT if you parameterize by $n-\chi$ S. Khot and V. Raman, ‘Parameterized Complexity of Finding Subgraphs with
Hereditary properties’.  If you parameterize by $\chi$ I would assume it's W[1]-hard.