In Frankenstein, which character's thoughts does the point of view allow readers to know?
The point of view is from that of Robert Walton, the ship's captain.
Mary Shelley chose to frame her bizarre story of Victor Frankenstein's creation of a living being with a more benign story of Robert Walton's attempt to unlock the mysteries of the Arctic, and by having such an epistolary style to her novel, Shelley created a narrative that seemed less shocking. In addition, Walton's letters introduce the idea of man's longings to conquer Nature as well as the concepts of loss and loneliness, thus preparing readers for Walton's discovery of Victor Frankenstein and his own tale of his attempts to master Nature. Also, Walton's search for someone with whom he can find friendship parallels that of the creature's longing for a companion.
That Walton is the Romantic and has found a soul of similar emotion is evinced in his letters at the end of the novel. For instance, in his letter of September 12th, he writes to his sister about Victor's speaking to the crew of the ship:
He spoke this with a voice so modulated to the different feelings expressed in his speech, with an eye so full of lofty design and heroism, that can you wonder that these men were moved?
Through the initial use of Walton's point of view, the reader of Frankenstein is also privy to the inner thoughts of Victor Frankenstein as he relates his personal history to the ship captain,and thus the creature, who later speaks to Victor. With this complexity of layers of narrative, Mary Shelley imitates the oral tradition of the ghost tale alluded to in the introduction of her novel.