The first reason why Frankenstein feels that he has the right to kill the Monster is that it is his creation. The second reason is that the Monster caused a lot of harm to him and his family.
Victor Frankenstein feels that he has the right to kill the Monster because he was the one who assembled it and brought it to life. When he first sees what he has created, he feels disgusted by its appearance abandoning the Monster. When they meet again, Victor’s primary goal is to escape from what he has done. Later, as the Monster starts to cause even more harm, Frankenstein feels responsible for its deeds. He believes that killing the Monster would save the world from what he has brought upon it.
But the character’s motivation is not only to care for the common good. Frankenstein hates his creation for personal reasons. At first, the Monster sought revenge and killed Frankenstein’s family. In the end, Frankenstein is driven by similar vindictive motives. When the Monster murders his wife, Elizabeth, and his father dies broken-hearted, he starts to chase him across Europe to kill his creation.