Pretty Steep Drive!
The route from Alamogordo NM to Cloudcroft NM on US 54 to US 82. In 19 miles, your elevation in Alamogordo starts at 4,400 ft, and ends at 8,700 ft in Cloudcroft. Average grade 5%. That is pretty steep, especially for such a sustained distance. You will use a lot more fuel during that climb, and if you have any concerns on engine over heating, you may want to do the drive before the heat of the middle of the day.
On the other hand, it should be a beautiful part of the country. Enjoy the trip!