We live on the east side of the city, but I worked next to Winton Woods county park for 30 years. They have a great campground and nice park facilities.
Winton Woods Campground - Great Parks of Hamilton County
Great reservation system, you can see each campsite on the reservation map.
Big Bone Lick SP in KY is an interesting spot. The campground is closed for the winter but we used to camp with our kids there when they were little. (Now in their 30's) Big Bone Lick