In general any two circles have two centers of homothety. They have only one center when the circles have the same radius or when the circles have the same center.
Given two circles of different radii with disjoint interiors, their external center of homothety is at the intersection of the external tangents and the internal center is at the intersection of the internal tangents. This is the case because homothety preserves tangency. So the centers of homothety are easy to construct in this case.
If the circles partially overlap, there is still an external homothetyic center at the intersection of external tangents, and an internal homothety center lying somewhere along the line joining the centers of the circles. If the circles overlap completlety (one inside the other), there are two internal homothety centers on the line joining the centers of the circles.
My question is how do you construct (with compass and straightedge) the centers of homothety in the cases where the circle interiors are not disjoint.