there are two cases of this problem. First is the intersection and second that is overlaping (containing).
First (intersection / polygon inside circle):
Find closest point on every edge of the polygon to the circle's center. If any distance between closest point to the center is less than radius, you got intersection or overlap.
Second (circle is whole in polygon):
Shoot ray from circle center to the right (or left/up/down) and count ray/segment (polygon edges) intersections. If intersection count is even circle is outside of polygon. If it's odd circle is inside.
I'll share picter from lectue for this case:
And take care of the singular cases.
Hope this will help.
edit: I think that it is just fair to add credits to the picture. Author is Petr Felkel, Assistant Professor at Czech Technical University in Prague