I am assuming when you say "go backcountry" that you are going backpacking, to carry large packs and camp down inside of the Canyon? For that you need a backpacking permit. If you already have your backpacking permit for the Hermit Trail, then the BCO (Backcountry Office) will give you the gate code to the road to Hermit's Rest. Trailhead parking is definitely a large area, but you need to ask BCO if you can park an RV there. You definitely cannot camp inside your RV there!!
Also, if you are going backpacking for three days, there is RV storage parking adjacent to the BCO, and from there you can take a shuttle bus that stops by the Hermit's Rest gate, and then take another shuttle bus from the gate out to Hermit's Rest visitor center. From there you will have to walk, I think it's less than 1/4 mile, to the trailhead, to start your backpacking trip.
If you were not talking about going backpacking, but just want to camp for free in the RV, you need to do that outside of the National Park. South of the village of Tusayan there are a number of dirt roads leading off into the National Forest. You can acquire a map for the Kaibab National Forest from the ranger station on the north side of Tusayan. Open weekdays, I presume. Anyway, you can drive down one of those dirt roads a couple miles and look for obvious pullouts used for camping sites. The map you want is the South Kaibab/Tusayan/Williams one. You can order it here: https://www.fs.usda.gov/main/kaibab/maps-pubs
Hope this is helpful.