On Crime, Punishment, and Fate
Raskolnikov was losing himself. Seriously ill, underfed, delusional, and alone. In the prior months he had forsaken his life by spending the little money he had left, leaving his occupation as a university student, and abandoning all of his relationships. The cause of Raskolnikov's downward spiral was a dire idea that was brewing in his head.
A sense of free will drives human beings. Knowing that you control your own actions and are able to choose your own course in life provides a feeling of deep purpose. That we can derive our own goals and whether or not we achieve them is primarily our own fault. This is why we get up every morning, to progress towards what we have decided is meaningful to achieve. It's just as important to know that if we don't get up the goal remains the same distance away. The idea of fate, meaning that no matter our actions we can not alter an inevitable trajectory in our own life, is the antithesis of free will and purpose.
Raskolnikov had started to believe that the life he was living was not his. Indeed he had lost the comfort of the idea of free will. It had been replaced with its grave antithesis, the belief that he was simply living passively and would resolve to the life intended for him by the world regardless of his actions. The world and everything in it except for him was in charge of his future. Feeling that everything happening to him was permanently out of his control inevitably drove him mad. In the depths of his psyche a grave response to this philosophy was brewing.
How does one respond to the terrifying idea of the absence of free will and the feeling of purpose it provides? It's terrifyingly simple: choose to commit an action that will inflict a permanent change on your life and environment. Refuse and take back your life by force. The more serious the act, the greater the extent to which you regain control of your life and environment. In his determination to take back his life, to decide his own fate and reach it through his willed actions Raskolnikov resolved to kill. He understood that deciding on and inflicting the death of another human is the deepest mark someone can leave on their environment and themselves. Permanently altering his own life and the life of another. He was determined that killing would propel him to human eminence, commanding not only his own fate but also that of the world around him.
"I... I wanted to dare, and I killed... I just wanted to dare"
The depths of his psyche had responded.
In the remaining pages of Crime and Punishment Raskolnikov suffers the psychological torment, remorse, and guilt that such an egregious act inflicts on its committer. Though Raskilnikov's deep misery leading to his eventual repentance and penance for his crime is the main subject of the novel, this thought of fate and passive living presented by Dostoevsky in the first pages is just as key. Free will and purpose lie at the core of the human psyche. Dostoevsky shows that even second guessing the actuality of the former can lead one down a dark path.
I'd like to note that there are many other themes in Crime and Punishment that are far more prominent than what I presented above. The rebuke of Nihilism and the importance of repentance for sins are perhaps the two main themes.