Ill start off by saying that you should probably have an AWD or 4WD high clearance vehicle for the upper campsites.  There are campsites lower on but are right next to the road.  Getting there you'll want to follow Dalton Wash Rd all the way to the plateau, staying left the whole way unless its private property.

There are no bathrooms or amenities here so either come prepared or take care of business before heading up or inbetween excursions.  The good thing from that is you'll be pretty secluded! (Cant speak for the peak months as we visited in May)  The upper campsites are ~45 minutes from Zion with the lower sites being only 30 minutes away!  Summer may not be the best time to visit as the sites are kind of exposed and theres not a ton of shade