How do I connect Frankenstein and Euthyphro to the question of who am I? What can I know? And how should we act?
In Plato's Euthyphro, Socrates questions him about the nature of piety. Euthyphro is a young man who begins the conversation certain of his religious convictions--certain enough to be prosecuting his own father for the murder of a slave (who had murdered someone else). Socrates ingratiates himself to Euthyphro by praising him for his wisdom while asking some rather pointed questions that amount to how Euthyphro can be so sure of himself. Specifically, how does he know what the gods want? What does it mean to be "pious"?
The paradox Socrates points out has become known as "Euthyphro's Dilemma," which essentially says (in Socrates's words): "The point which I should first wish to understand is whether the pious or holy is beloved by the gods because it is holy, or holy because it is beloved of the gods."
In simpler terms, we might ask the equivalent: "Is something good because God desires it or does God desire it because it is good?" Socrates also points out that both cannot be simultaneously true and amounts to circular reasoning (in Socrates's words, "The argument, as you will perceive, comes round to the same point"). To wit: If God desires something because it is good and it's good because God desires it, we've arrived right back where we've started.
Forced to choose, then, we encounter problems either way (this was also true in Socrates's day). If God desires something because it is good, then that means morality exists independently of God and God's omniscience is limited, since he does not control morality. If, on the other hand, something is good because God desires it, that makes morality arbitrary and whatever God does or commands is good by definition (including genocide, human sacrifice, etc.).
Thus, this is essentially a question regarding how we can know right from wrong.
Frankenstein's monster struggles with this problem--how to know right from wrong, and why something is right or wrong--himself. Consider that Victor Frankenstein is his creator, so in a sense, is his god. His god, however, has not given him a moral code and the monster goes through life trying to work out for himself what is right and what is wrong, and why. He initially bases his idea of "right" and "wrong" on watching the people around him. He tells his maker that he was driven from a village by the "barbarous villagers" and he resolves to not behave that way; by the same token, when he realizes that the family he watches are impoverished, yet themselves go hungry to ensure the old man could eat, he says, "This trait of kindness moved me sensibly. I had been accustomed, during the night, to steal a part of their store for my own consumption; but when I found that in doing this I inflicted pain on the cottagers, I abstained, and satisfied myself with berries, nuts, and roots."
Thus, he works out for himself what is right and wrong without the help of his "creator," who himself is riddled with moral shortcomings (Victor agonizes over having "played god" and created life, and later, when his creation turns to anger and violence, feels himself responsible for its murders). In this way, Frankenstein mirrors the dilemma Euthyphro himself faces: Who should the monster turn to to know what "piety" is? Is his "god" good because he's a god, or is he a god because he's good (which he actually isn't, as we see, but the same can be said for any other god)?
As to the question of who you are, how you should act, and what can we know.... Each person must answer these questions for himself. If you discard Euthyphro's dilemma, you may embrace your idea of the divine for answers. Otherwise, you will find that you must turn to your own knowledge and logic to discern them.