If you come to North Carolina, I do not recommend Boone, Blowing Rock, or Beech Mountain (Basically above Banner Elk). There really isn't any place to train around these areas save Moses Cone Park, which has about 20+miles of dirt roads, some of which will take you up as high as 4400-4500ft, so good climbing as far as % grade goes. I lived in Boone area for 3 years. It's too crowded now, nothing but the university (AppState) and the tourists. That's about it. Traffic is worse than Charlotte most days. If you like the crowds and chasing horse s**t on trails, go for it. ZAP Fitness advertises like Blowing Rock is so awesome, but it's not, not really. Just my humble opinion, but I lived in the area and built ski cabins near Appalachian Ski Mountain so my opinion is based on experience.
If you want solitude and good trails then you have to utilize the middle and southern portions of the Pisgah National Forest which covers about 500,000 acres and of which there are like 400+ miles of National Forest Service access roads and trails that are more for hiking but in good enough shape for running. Some single track bike trails that could be used as well. Some climb up to the Blue Ridge Parkway, so you'd be getting up around 4500+ feet. If you go the Black Mtn ranges, you'll come across Mt. Mitchell, highest point east of the Mississippi at about 6700 feet. There are a few trails in the area you could use, mostly single track mountain bike, but you might could use them. Down around the Brevard area, going west towards Cashiers, Highlands, and Franklin there are even more, again, some of which climb up from the area (2500ft or so) up to the Blue Ridge Parkway (4500+ through that area).
The posts about altitude training are right. You'd need a minimum of several weeks up above several thousand feet. But even if you did all summer at 6 or 7K feet...when you moved back home, within a few weeks you'd lose any gains. You're body responds to O2 increase or decrease by generating/removing EPO which tells the red marrow in your long bones to make more red blood cells and the hemoglobin protein in them (carries the O2). So after even a few days at altitude your body does begin to respond but it takes time, usually 3 weeks, before your body adapts to altitude stress. But if you come down from altitude the reverse happens. Your body will get rid of those extra RBC's since there is more oxygen closer to sea level (say below 2K feet or so). But the thing you will get is strength from climbing steep or long percent grades. NC has a lot of roads that lead to the Parkway or up minor mountain peaks that climb several thousand feet over just a few miles. I've run plenty of them myself back in the day and they hurt! But they'll make you strong! Good luck!