Well here's a start...
I've changed a few bits and pieces to make life easier for myself, like ignoring the Distance struct which I didn't see the point of, and changing the declaration of sumRadius(). You also need to decide whether or not you consider circles that touch as overlapping. Anyway, hopefully this will get you started...
#define MAX_CIRCLES 100
double centreDist(struct Circle circle1, struct Circle circle2);
double sumRadius(struct Circle circle1, struct Circle circle2);
void getCircle(struct Circle * pCircle);
struct Circle circle[MAX_CIRCLES];
unsigned int i, j;
for (i = 0; i < MAX_CIRCLES; i++)
printf("Enter Circle #%d: ", i);
overlap = false;
for (j = 0; j < i; j++)
if (centreDist(circle[i], circle[j]) <= sumRadius(circle[i], circle[j]))
overlap = true;