The Alaska flight mentioned in the article was a special invitation-only event. The only way for a member of the public to get a seat on it was to win a contest on the airline's social media.
There are a number of eclipse-chasing clubs around the world, and from time to time they may organize a charter flight to catch an eclipse from the air. Joining one of these clubs, or at least their mailing list, would probably be your best bet for seeing an eclipse from the air, as a charter would have the least uncertainty, and you could expect the flight path of such a flight to maximize your time in the umbra and have ample opportunity to view it from either side of the plane. According to the linked article, a window seat cost as much as US$8,500.
As noted on Alaska Airlines' blog post about the flight, there are regularly scheduled flights that go through the path of a solar eclipse, as another Alaska flight in March 2016. As Space.com columnist Joe Rao advises, you could compare the predicted path of the eclipse with flight tracks on sites light FlightAware.
Glenn Schneider, an astronomer at the University of Arizona's Steward Observatory and noted umbraphile, has developed a piece of software called EFlight which helps calculate what flights to aim for, although it is for serious enthusiasts and is not a point-and-click affair. He used this software to recommend a change in the flight plan for the 2016 Alaska flight.
Naturally, scheduled commercial flights are more constrained in their flight paths and and the time they have to linger, and there is always a chance you would be routed out of view. Still, if you don't have $8500 to spare hoping an event that lasts two or three minutes doesn't get clouded out, the commercial flights seem to be a reasonable bet.