Last August, in between performances of Joseph and the Amazing Technicolor Dreamcoat at a summer-stock theater in Maine, Clay Aiken studied the Democratic Congressional Campaign Committee’s demographic breakdown of North Carolina’s second district. He concluded that, although challenging, the race was winnable. By the time he returned to his 9,000-square-foot mansion outside Raleigh that fall, the flame-haired former “American Idol” star was ready to run.