How does the concept of hubris figure into the plot of Frankenstein? Are the characters propelled by fate or something more psychological?
Victor's hubris certainly blinds him when he is working on the creature. He says,
A new species would bless me as its creator and source; many happy and excellent natures would owe their being to me. No father could claim the gratitude of his child so completely as I should deserve theirs.
If this isn't playing God, then I don't know what is. Not only does he try to make a human, but he attempts to improve upon what God has already done: he makes his creature eight feet tall and purposely selects parts that he thinks will be physically beautiful. He makes a superhuman, unlike anything God ever made. His proud claim that no father could claim his child's gratitude as much as he would be able to claim his "children's" seems to invite punishment.
Moreover, Victor's proud inability to take responsibility for his errors or even admit wrongdoing leads to his downfall. Despite the fact that he gives Captain Walton all kinds of sage advice about knowing one's limits and learning to appreciate one's hometown and not seeking to know more and go further, when Victor nears death, he tells Walton, "During these last days I have been occupied in examining my past conduct; nor do I find it blamable. In a fit of enthusiastic madness I created a rational creature and was bound towards him to assure, as far as was in my power, his happiness and well-being."
He goes on to discuss why he had to refuse the creature a companion, and that's all well and good, but he seems to conveniently forget that there were other things he could have done, such as not abandoning the creature because he was ugly, opening the creature up to a lifetime of horrid loneliness and isolation. But Victor is too proud to take any responsibility.
Hubris (extreme arrogance) is, of course, one of Frankenstein's major themes. If it were not for Victor's tragic flaw of hubris, the book's plot would be completely different, for Victor states that his goal in creating another human is to bring glory to himself. He desperately wants to be recognized for his accomplishment. Likewise, his refusal to admit the truth to others who could actually help him subdue the Monster stems from his belief that he can handle the situation himself and his pride in his reputation with his family and friends.
Walton, the novel's original narrator, is guilty of hubris as well. He travels to the North Pole so that he can bring back to mankind some unknown information such as the cause/source of the earth's magnetism. His sister's uneasiness with his pursuit demonstrates that she recognizes her brother's flaw, as does Victor when he meets Walton and sees in him the same desperation for glory that he himself once possessed.
In answer to the second part of your question, Shelley wrote the novel as a typical Gothic/Romantic work; so the element of fatalism does not play much of a role. It is associated more with the Naturalistic tendencies of later writers. Additionally, throughout much of the novel, Victor and other characters have choices. Each choice could have changed the outcome of many lives; so a well-supported argument that fate drives Frankenstein's characters as it does in a work such as Tess of the D'Urbevilles by Thomas Hardy would be a difficult task.
I have a slightly different take on the hubris issue. Victor and Walton's behavior always struck me as more obsessive than proud. Both are seeking for a "truth" that takes over their entire lives. In Victor's it leads him to isolation from his family and from all the more important passions (remember that although this is a Gothic novel, it also has many elements of the Romantic period which would place a higher emphasis on knowledge gained through intuition/passion as opposed to that which comes from scientific knowledge, Victor's obsession.
And in the end, it is not Victor's pride which leads to his downfall, but it is his inability to FEEL for his creation; instead of loving what he had created, giving the creature the affection he so craves throughout the entire story, he rejects it from the moment of its "birth." It continues to find this rejection throughout its entire life.
Perhaps you could call this hubris, but I think it is something quite different. Victor wanted to make "the" great contribution to humanity, to defeat death. There may be hubris in that, but we continue to try to this very day ....