Realism began in the 19th century during the Victorian era as a reaction to the sentimentality of Romanticism. Realism was dedicated to expressing detailed passages of accurate observation without the symbolism and emblems of the Romantics. Realism had a unique relationship with journalism: fiction was looked at as reporting occurrences with objectivity. Part of what was fictionally reported were the stories of lower classes who suffered disadvantages and neglect and the new middle class that continued to struggle to secure its place in society.
Characters were given new importance as plot became less important and individual motivation became more important. The new science of psychology popularized the concept of subconscious and introduced the debate about the roles of nurture and nature. Realist writers incorporated these concepts and debates into their novels along with the tenet that humans are neither good nor bad, but rather a mixture of both.
Regarding style, Dostoevsky certainly reports every minute detail of Raskolnikov's journey and surroundings. To this extent, this novel is a perfect example of Realism. An example is the detail in Raskolnikov's journey to the old woman's room to explore carrying out his deed. Every detail is told, including the number of steps he walks and the style of his hat:
He had not far to go; he knew indeed how many steps it was from the gate of his lodging house: exactly seven hundred and thirty.
It was a tall round hat from Zimmerman's, but completely worn out, rusty with age, all torn and bespattered, brimless and bent on one side in a most unseemly fashion.
Regarding character, the primary interest in the character of Raskolnikov, the quality that makes him live immortally in the Realist tradition, is the detailed descriptions of his psychological state, his thoughts, and his motivations. In fact, Raskolnikov is numbered among some of the most lauded Realist characters ever written. We are told his psychological battles; his feelings; and the motivations behind his actions. Raskolnikov drives the novel instead of plot and action driving it. This quote shows his thought process, his rationalization, his judgements ("ought"), and his decision with the motivation ("spoil the whole plan"; "makes it noticeable"):
"I knew it," he muttered in confusion, "I thought so! That's the worst of all! Why, a stupid thing like this, the most trivial detail might spoil the whole plan. Yes, my hat is too noticeable.... It looks absurd and that makes it noticeable.... With my rags I ought to wear a cap, any sort of old pancake, but not this grotesque thing."
In sum, Crime and Punishment, written in 1866, is critically considered an exemplary representation of Realist literature in terms of both style and character.