I'm not any expert in medicine, but some of the best pre-med programs to go into include Johns Hopkins in Maryland, Duke University in North Carolina and Stanford University in California. (According to Google, anyway)
I'd assume if you're going into medicine you have rather high grades, and depending on where your cousin is located I'd take that into consideration when you apply to a med school in a certain state. You apply to each university separately on their website and your college essay must have a heavy emphasis on what you can contribute to the school community (any past activities you contributed to are important here) and then you'll need to get a VISA which is easy enough if you live in the UK.
Hope this helps