"Barely ran high school" is pretty much the story. Although he ran cross-country starting as a freshman, on the whole he was either absent or unremarkable as a distance runner until his senior year.
He was 44th at the NH XC Meet of Champs as a sophomore. He finished 13th (just ahead of Hanover High School's 5th, 6th and 7th runners) at the NH Class I State Meet as a junior and evidently didn't run the MOC at all. I don't think he even ran track until his senior year.
In his final season of high-school cross-country he exploded into prominence, repeatedly turning away Aaron Watanabe, who came into the season as the overwhelming favorite to go unbeaten and who the next spring would become the first NH runner under 9:00 in the 3200. But he didn't make much of an impact beyond his home state's borders, finishing just inside the top 20 in both the New England XC Champs and the Foot Locker Northeast race. Then in track he tripled at his indoor state meet, winning the 1500m and 3000mand coming in second in the 1000m. He grabbed fifth the the two-mile at the New Englands that winter with a 9:24.66. He won the outdoor MOC in 4:15.
So by the time he was done he was plenty good, at least to the extent that as NH kid he didn't have to run blazing fast to win almost everything. Portsmouth, NH was not, is not, and most likely never will be a distance-running powerhouse (although Cory Thorne went there) and it isn't much of a reach to assume his training before college was pretty much a joke.
He appears to have redshirted cross-country in 2012, which makes sense as NU is a five-year school.
All in all, what people are seeing now is the continuation of a rapid steady progression that started just over three years ago, and he won't have to improve all that much to establish himself as the fastest guy ever from NH in the middle distances.
I was in high school in new hampshire when jenkins was (a few years younger than him). I can confirm much of what this post says. However, I should point out something about your last sentence. Russell Brown went to high school in new hampshire and has run 3:34 and 3:51 for the 1500m/mile. So while Jenkins is one of three new hampshire citizens to run a sub 4 minute mile, he still has a ways to go to become the fastest.
I look forward to seeing what he does this spring. His freshman year he was the only true freshman to make it to NCAA outdoors in the 1500 I believe.