Natty is 100% correct. Using organics in turf and landscape is about building the quality of soil.
It sounds to me like this property has already been hit with synthetic fertilizer and pesticides. Because you said it was chuck full of clover and other weeds, I suggest that you 1st try to gain an upper hand on these weeds with IPM applications of selective broadleaf herbicide.
Q) How do you keep weeds like clover from re-entering a lawn?
A) Whether or not your lawn care preference is organic, the first line of defense against weeds in any lawn is to establish a stand of healthy, dense turf. Which leads to the next question...
Q) Is May / June typically the best time to renovate / overseed thin lawns?
A) NO. The best months of the year for seeding are usually Aug / Sep. The reason for this is warmer soil temperatures coming out of the summer months typically make for more efficient seed germination.
1) Commence organic fertilizer program immediately.
2) Focus on broadleaf control with limited use of synthetics (IPM).
3) With seeding (preferably in fall) thicken the density of the turf to the point that the turf itself becomes its own primary weed control mechanism.
4) Aeration 1x - 2x annually, spring / fall, never summer or winter