A problem similar to the Bin packing problem?
One way to solve your problem is to use binary search over the maximum weight.  In other words, pick a threshold $t$, and check whether it's possible to pack your items into $m$ bins, without using more than a total weight of $t$ in each bin.  This is exactly the bin packing problem: is it possible to pack the items into $m$ bins of capacity $t$?  You can apply any algorithm for bin packing to this problem.  Then, use binary search on $t$.

Be warned that it is NP-hard, so don't expect an efficient algorithm that produces the optimal solution in a reasonable amount of time for large problem instances.