Hello:
I am a U.S. Citizen, but am able to apply for a British passport, as my parents are British citizens. Will I still require a student Visa before I start a postgrad program in the UK this fall?

You wouldn't need a visa if you have a British passport, since you would be considered a British citizen. However, since you have not been living in the UK for the last few years you would be paying overseas tuition.
The school you will be attending will be able to help you out with all that is required as well.