My understanding is that, at this time, "Scottish citizenship" does not exist, because Scotland is not an independent state [at least not yet]. Your father would have been a British citizen, and as his daughter, you are probably already a British citizen by birth. IF you were not born out of wedlock. I keep hearing that British citizenship processes are a pain in the neck, but provided you can prove a) your father's citizenship, b) your parents being married at the time of your birth, and c) that you are the daughter of your father, you should be able to register with the UK and gain official status as a UK citizen. http://en.wikipedia.org/wiki/British_nationality_law#Before_1983
I have no idea what UK office that would be, though, or who exactly you'd have to talk to. I would start by calling up or emailing the British embassy in D.C. or one of the consulates. http://ukinusa.fco.gov.uk/en/about-us/other-locations/
You may also need their help obtaining proof of your father's British citizenship, if you do not have his old passport/paperwork/etc.
My only source for this information is wikipedia, however, and I noticed that some of the information there is incomplete. There IS the possibility that your father lost his British citizenship when he emigrated, if he then gained U.S. citizenship... that, unfortunately, is not specified in the wiki here, but it was not uncommon for states to revoke citizenship if someone left and gained citizenship elsewhere, to prevent conflicts of loyalty, etc. These days, dual or multiple citizenship is acceptable in most if not all English-speaking countries, but that's the result of a lot of legislation between WWII and the early 90's. If your father _did_ become a U.S. citizen after he came over, that's definitely something to ask about before you go through all the paperwork.