Lower Karr Canyon campground is about 6 miles from the main highway at High Rolls, NM. The last 2 miles or so being rather rough dirt. Most cars should be able to make it if driven with care, though.
The campsite itself is rustic, no amenities other than the pit toilet which is just fine. The sites are on the other side of a closed bridge, so you gotta carry your stuff a quarter of a mile or so for the sites furthest away from the parking area. Also, being deep in the canyon means that there's no cell signal unless you hoof it a fair bit up a hill.
It's a very secluded experience despite being so close to Cloudcroft/Alamogordo. Highly recommended if you want a no-frills campsite that will make you work a bit to get to it and then some more to settle in.