Near the end of Chapter 16 of Frankenstein, the Monster tells Victor:
"The sleeper stirred; a thrill of terror ran through me. Should she indeed awake, and see me, and curse me, and denounce the murderer? Thus would she assuredly act, if her darkened eyes opened and she beheld me. The thought was madness; it stirred the fiend within me--not I, but she shall suffer: the murder I have committed because I am for ever robbed of all that she could give me, she shall atone. The crime had its source in her: be hers the punishment! Thanks to the lessons of Felix and the sanguinary laws of man, I had learned now to work mischief. I bent over her, and placed the portrait securely in one of the folds of her dress. She moved again, and I fled.
The Monster does not feel remorse for his crimes. In fact, he justifies them according to the "sanguinary (blood) laws of man." The Monster thanks Felix for the lessons he has taught him. According to a University of Penn professor:
A particularly brutal aspect of the "sanguinary laws of man" that the Creature did not learn from Felix but has discovered on his own is how to victimize women.
Not only does Shelley characterize the Monster as a walking, talking embodiment of Revenge, but his also the embodiment of Sexism. Since women were second-class citizens, the Monster has learned from men not to care about the plight of the weaker sex.
In the final chapter of Frankenstein, the creature does express remorse for his heinous acts because they have caused the death of his creator. Certainly, he weeps over the body of Victor Frankenstein, whom he has loved from the moment he opened his eys, displaying more feeling for Victor than Victor has had for him. As he regards the body of his creator, the creature is filled with the "wildest rage of some uncontrollable passion":
'That is also my victim!...in his murder my crimes are consummated....Oh, Frankenstein! generous and self-devoted being! what does it avail that I now ask thee to pardon me?'
He tells Walton,
But it is true that I am a wretch. I have murdered the lovely and the helpless...I have devoted my creator, the select specimen of all that is worthy of love and admiration among men, to misery....your abhorrence cannot equal that with which I regard myself.
Then, as he looks again upon Victor, the creature speaks of his "bitterest remorse" and tells his creator that the desire Victor has had to seek revenge is not as great as the desire for vengeance against himself that he feels:
Blasted as thou wert, my agony was still superior to thine; for the bitter sting of remorse will not cease to rankle in my wounds until death shall close them for ever.
Despite his having exacted revenge against Victor, the creature feels sorry for his creator and rues that he has destroyed the man he loved in destroying all that Victor loved. Indeed, there is a plea for forgiveness in his awful remorse.
I assume that you mean an apology like Socrates's apology -- one where someone is explaining themselves, not where they are saying they are sorry.
I would say that the monster committed the murders because of how badly he was treated. I would say that he was driven to it by the fact that he was rejected by all the people he met simply because of the way he looked. He really was not completely to blame because he was just acting out of frustration because of how his maker had made him.