Modified Bellman Ford to find minmum cost cycle in O(E²V) time?
Some hints for you, so you can solve your own exercise:


If the graph has a negative-weight cycle, what is the proper answer?  Can you detect this situation?
If the minimum-weight cycle has positive weight (strictly greater than zero), how could you find it?  If you knew the distance between every pair of vertices, could you use that to help you find the minimum-weight cycle in this case?  Does that suggest an algorithm that could handle this case?
If the minimum-weight cycle has zero weight, can you detect this fact?  What could you do in this situation?