Sounds like a really fun trip! The Painted Hills are one of my favorite places in Oregon, with a landscape that is both majestic and a little surreal when the sunlight hits it just right. One of the best campgrounds near the Painted Hills is the Clyde Holliday State Recreation Site, located between John Day and Mount Vernon right along the cool waters of the John Day River. This campground has both restrooms and hot showers available, and it has easy access for vehicles. The Mitchell City Park nearby also has restrooms but is in the town itself. To the west of the Painted Hills there is camping at Ochoco Divide that has restrooms and easy car access as well. The majority of other campgrounds in the area are considered primitive with less amenities.
The option for campfires in most places in Central and Eastern Oregon in August will be dependent on the posted level of fire danger, which at that time of year is likely to be high. Also keep in mind that many campgrounds in the region are already reserved for events surrounding the eclipse in later August, so try and beat the crowds by planning your trip early.
Information for the State Recreation Site is available here. Information for the Ochoco Divide Campground is available here. The National Park Service also has some helpful information about camping, information about the Painted Hills, and other services here.