One of the nicest campgrounds in the area is the KOA in Streetsboro (near Kent) and it would be about 30-40 minute drive into Cleveland depending on where you are going. Add a few minutes if during rush hour. Another option is Punderson State Park, which is east of Cleveland. Probably about the same commute time. And, American Wilderness Campground (Grafton, OH) wouldn't be a bad location...probably a better commute than Streetsboro or Punderson as you could just go north on 83, pick up the 480 and then the 71 into downtown Cleveland. I suggest you check mileage and travel time to where you have to be in Cleveland using Mapquest.com. We've stayed at Streetsboro many times, and we've been to Punderson and American Wilderness one time each.
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