As a representative figure of white society, Lula is steeped in ignorance and prejudice. She knows nothing at all about African Americans and their culture, and her view of them has been built out of crude racial stereotypes handed down from generation to generation. As such, Lula is unable to talk to Clay as if he were another human being; she relates to him purely as a stereotype. Almost every time she opens her mouth, she reveals her invincible ignorance as well as her underlying contempt for black people.
As an allegory of American race relations, Dutchman presents us with a pretty grim picture of the mutual incomprehension that exists between black and white. White America, as represented by Lula, clings tenaciously to hard and fast stereotypes about African Americans. And when such stereotypes are challenged, as they are in the play by the well-dressed Clay in his Ivy League suit, the instinctive response is fear and loathing.
Like the white society she represents, Lula uses racial stereotypes as a means of keeping African Americans in a state of subjection. One such stereotype is that black people are deferent toward white people, their alleged racial superiors. So when Clay, after Lula's insistent goading, finally reveals his antipathy toward white people in a splenetic outburst, Lula stabs him to death. So long as she could manipulate him, she always felt in control. But now that Clay's got out of his box, so to speak, Lula puts him right back there by killing him.