Graph coloring with fixed-size color classes
This problem is NP-hard: it is at least as hard as independent set.  In particular, if you want to know whether there exists an independent set of size $N$, ask for a coloring with as many colors of size $N$; if you find any coloring where a single color occurs $N$ times, you know there's an independent set of size $N$.  So, you should not expect any efficient algorithm for this problem that works on all problem instances.
One plausible approach is to use a ILP solver.  You can define zero-or-one variables $x_{v,c}$, where $x_{v,c}=1$ means that vertex $v$ is assigned color $c$.  Then it is easy to express the requirement that a color $c$ be assigned to exactly $N$ vertices: we require $\sum_{v \in V} x_{v,c} = N$.  The constraint that two adjacent vertices $v,w$ be assigned different colors can be expressed by $x_{v,c} + x_{w,c} \le 1$ for all $c$, and that each vertex $v$ receive a color by $\sum_c x_{v,c} = 1$.  Without loss of generality, to test whether it is possible to have $k$ colors be assigned to $N$ vertices, you can constrain the first $k$ colors to have $N$ vertices and put no constraints on the remaining colors.  Then, use binary search on $k$ to find the largest $k$ for which a solution exists.
Another plausible approach is to use a SAT solver.  You could define variables $x_{v,c}$, where if $x_{v,c}$ is true then vertex $v$ is assigned color $c$, and express your constraints in SAT.  You can require that color $c$ be assigned to exactly $N$ vertices, by requiring that $N$ out of the variables $x_{\cdot,c}$ are set to true (see Encoding 1-out-of-n constraint for SAT solvers and links for methods).  Otherwise, this is similar to using an ILP solver.
These might work if $N$ is small enough and the graph is small enough, but eventually will run into exponential behavior once the problem gets large enough.