Elena Gilbert's doppelgänger, Katherine Pierce (Nina Dobrev), is one of the biggest villains on The Vampire Diaries, and she runs from Niklaus "Klaus" Mikaelson (Joseph Morgan) for a serious reason. Katherine has a dark history with Stefan (Paul Wesley) and Damon Salvatore (Ian Somerhalder), as she knew them in Mystic Falls in 1864. After appearing in the Gilbert home in The Vampire Diaries season 1, episode 22, "Founder's Day," she begins her tradition of pretending to be Elena. Throughout her arc, Katherine has many enemies, including Elena, whom she's envious of and wants dead, and Stefan, who doesn't appreciate her cruel nature.
https://screenrant.com/vampire-diaries-katherine-klaus-500-year-rivalry-explained/