The creature tries to appeal to Victor's emotions as well as his sense of responsibility as a creator. The creature says,
I am thy creature, and I will be even mild and docile to my natural lord and king if thou wilt also perform thy part, the which thou owest me. Oh, Frankenstein, be not equitable to every other and trample upon me alone, to whom thy justice, and even thy clemency and affection, is most due. Remember that I am thy creature; I ought to be thy Adam, but I am rather the fallen angel, whom thou drivest from joy for no misdeed. Everywhere I see bliss, from which I alone am irrevocably excluded.
In other words, the creature describes himself as a victim. He implies that Victor, his creator, has dealt fairly with everyone else in his life except the creature. The creature claims that Victor owes him the chance to be happy. In fact, he claims that Victor owes him mercy and affection because he is his maker. He appeals to Victor as to a god, perhaps stroking his ego a bit, saying that he should be Victor's prized and loved creation, but he is, instead, punished and ostracized for no fault of his own. He appeals to Victor's sense of justice, describing the happiness everyone around him can share but of which he, alone, can never partake. He desires a companion to alleviate the pain of solitude.
Further, the creature describes the horrible treatment he's received from all of humanity. He says, "These bleak skies I hail, for they are kinder to me than your fellow beings." He knows that he will never be accepted by humanity, due to his horrible appearance; therefore, he must have a mate as physically ugly as he. He says, "I am alone and miserable; man will not associate with me; but one as deformed and horrible as myself would not deny herself to me. My companion must be of the same species and have the same defects." A mate will cure his loneliness, and he believes that Victor owes him at least a chance to be happy.