According to genealogist Patrick Cracroft-Brennan, Kate and William are very very (very) distantly related. It comes as little shock to some, as millions of people do indeed have ties to the Royal Family in some way. According to the U.K.'s Channel 4, Patrick explained Kate's familial connection to William as being "fourteenth cousins once removed through his mother and fifteenth cousins through his father."
He explained, "Her great-great-grandmother, Frances Elizabeth Greenhow, was the 10 [times] great-granddaughter of Sir William Gascoigne, a Yorkshire knight who died in 1487. He married Lady Margaret Percy, 4th and youngest daughter of Henry Percy, Earl of Northumberland. The Earl descended from King Edward III through both his parents. [King] Charles and the late Princess Diana both descend from Sir William Gascoigne and his wife Lady Margaret."