http://en.wikipedia.org/wiki/Stephen_Uro%C5%A1_IV_Du%C5%A1an_of_Serbia
called himself the emperor of serbs and greeks and used byzantine symbols such as the double headed eagle
as for the wars they were war for the domination of the region. Don't put together the ambitions of the overlord and the relationship between the peoples
Because you have a serbian empire, a byz empire and a bulgarian empire competing for the domination of the region doesn't mean that on the long term the people there wouldn't be okay no matter who is the one ruling, the culture were close enough
Serbian king claimed the title of Byzantine emperor so that he could justify his conquest of Orthodox Greek lands held by Catholic crusaders.
However, that particular case is rather complicated. Stefan Dusan was expelled from Serbian court by his father, so him and his Bulgarian mother (iirc) basically lived at - Constantinople - until he inherited the throne. That guy was basically more Greek and Bulgarian than Serbian (oh the irony) simply by events in his life.
But with all that said, Serbs never claimed to be Romans, youre just wrong on that.