The Endurance Challenge is enabled in the Abstergo Challenges Cheats menu, and requires that you have completed at least 55 challenges.
You can get this while you are getting:
Incapacitate the vessels, start the boarding process and use the Puckle Gun to kill off enemies on the other ship. Both French and Assassin ships/enemies count. I was on my fourth French Brig when it popped, with about 37 total enemies killed. The extra seven were possibly because of enemies that were killed by my crew?
The easiest ships are found in the southwest area of North Atlantic.