While Shahrazad is indeed trying to stay alive through her storytelling, she has a greater purpose for telling stories to Shahrayar. What are two outcomes she is trying to achieve for Shahrayar through her storytelling? How do some of the stories relate to the frame of One Thousand and One Nights and help her achieve this end?

By telling stories to Shahrayer, Shahrazad is putting her virtuosic mastery as a storyteller to the most dramatic test imaginable. She is also making an ambitious claim to rule, which succeeds when her inventiveness, statecraft, courage, and practical wisdom pay off and she is made royal consort, in addition to saving her sister's life.
In other words, Shahrazad's story is, in effect, a handbook for would-be rulers and public servants. Its implicit message is that mastery in the art of storytelling amounts to a plausible claim to rule, because it entails mastery of so many of the virtues appropriate to a ruler.
All of the stories told by Shahrazad can be read with reference to their conveyance of the message that the steadfast practice of the virtues—even in the face of duplicitous treatment by others—lead to the conferral of reward and responsibility. Thus, they contain a guidebook for rulers to follow (i.e., the factors that should be considered when elevating others to high positions).

In telling her story Shahrazad is also trying to save the life of any woman unfortunate enough to cross paths with the Sultan. The Sultan is hell-bent on exacting revenge upon the whole of womankind and desperately needs to be stopped before he can do any more damage. So Shahrazad bravely presents the murderous Sultan with a rounded portrait of humanity, especially of the female of the species, which allows him to see a side of people to which he's previously been blind.
In "The Tale of Sympathy the Learned," for example, the Sultan is presented with a picture of a slave girl, who despite her humble origins and despite her restricted gender role is nonetheless highly educated, with extensive knowledge in a variety of fields such as law, poetry, astronomy, and arithmetic. What Shahrazad is trying to do here is to show the Sultan, ever so subtly, that women can be intelligent, trustworthy, and above all, supremely virtuous.

While she is undeniably trying to stay alive by enthralling the king with her stories, Sharazad’s larger purpose is to serve as a kind of therapist. The king has been driven insane by the betrayal of his wife and has become a sexual predator and murderer—each night he beds a different virgin, and the following morning he executes them. The stories Shahrazad tells are designed, in the first place, to cure the king’s depravity.
A second outcome Shahrazad is trying to bring about is to educate the king and make him a more humane person. Shahrazad’s tales, as Sadhana Naithani points out, are deeply humanistic; their use of fantasy notwithstanding, they are “stories about human beings and their relationships” in balance with nature. Shahrazad’s stories show human nature to be “a reflection of Mother Nature herself”: uncontrollable, prone to both contradictory actions and emotions, yet ultimately without malice or indeed any specific aim. This kind of balance is the first step in the king finding his “own place in the scheme of things.”
Shahrazad’s role as a kind of stealth teacher inverts the patriarchal power structure. The king is powerful and destructive, but the real hero of the Arabian Nights is Shahrazad. She is exceptionally learned and a tremendous storyteller, and she is extremely confident in her power to teach the king to be more humane.