Was at the Grand Canyon both rims last October. September and October are glorious times to be there.
For the North Rim and FHU, choices are limited. Kaibab CamperVillage at Jacob Lake is the only place I recall that has FHU. There's also Jacob Lake CG, it's USFS CG with no hookups. Jacob Lake is 45 miles from the North Rim. Within the North Rim there's the CG within the park, which is fantastic, but no hookups. Outside the park, closer to the entrance, there's Demotte USFS CG, again no hookups. And there's boondocking areas between Jacob Lake and the park entrance.
For the South Rim, Trailer Village is the logical choice. Inside the park, FHU, convenient to everything to see and do on that side of the canyon. Shuttle stop at the entrance to Trailer Village, takes you everywhere you want to go...for free.
Drive around from North to South is 197 miles. Plan to stop at Navajo Bridge for some great photo ops and some great history of the area. Allow for about 5 hours to get around.