Is it possible that I can get a Duke University user account to access math videos?
As a commentor wrote, "ask Duke". In practice, that means the following.

Note from the URL that the page is maintained by the math department. If you can, try to figure out who specifically is maintaining the webpage (i.e., a person's name), and contact them. Otherwise, I'd recommend writing to anyone in a position of responsilibity in their math department -- say, the graduate director. Tell them that you are interested in watching their videos, and ask if there's any way you could be permitted to do so.

The most likely outcome is either that they say no, or that you get no response at all -- but in my estimation it's okay to ask, and they might say yes.