The first answer you've received covers much of what I would say; however, in the movie, Kate is the victim of a bet--pure and simple. She is "shrewish" because she's been burned before and when she finds out about the bet, she comes unhinged (understandably so). Bianca is the one everyone wants in the play, and while there is no "bet" placed in the actual play, Kate stands in the way of all of Bianca's suitors. The eldest daughter must marry before the youngest or face a life of spinsterhood. Therefore, Bianca's suitors attempt to arrange for Kate's marriage as a means to get to Bianca.