I received Royalty distributions from BP Prudhoe Bay Royalty Trust UBI. How do I properly claim these as I don't own then property nor a business?

Yes, that information can be entered in TurboTax. 

I assuming you received a 1099-MISC reporting this information. If you received a Schedule K-1, please respond back to this thread as the directions are different. 

This is where to start:

1. Select Federal from the left side menu.
2. Select Wages & Income.
3. Scroll down and look for Rental Properties and Royalties (Sch E)
4. Click Review on the right.
5. Click Edit to the right of your 1099-MISC entry, or answer Yes, if you haven't entered anything yet.
6. Select Royalty, click Continue.
7. Fill in any missing information and Continue.
8. On the screen, What type of royalty income is (name)? choose Oil and gas income.
9. Continue through the questions.
10. On the screen What additional expenses and assets did you have for (name)? you can indicate if your have any expenses. 
11. Your Depletion will calculate automatically and if you have any fees or taxes paid, you can deduct those toward the end of the section. The interview does refer to this as a business, but this is the correct section for royalties.