Raskolnikov, an intense and deeply confused young man, represents in microcosm the paradoxes of the urban liberal society then just starting to take shape in Imperial Russia. On the one hand, Raskolnikov is fiercely individualistic, believing that the true locus of value in society is the creative individual. Such individuals—strong, single-minded, and utterly ruthless in getting what they want—are not subject to the petty restrictions of bourgeois law and morality. And it is primarily in such proto-Nietzschean terms that he justifies murdering the old pawnbroker.

At the same time, Raskolnikov claims that by destroying this supposedly inferior specimen, he will be doing society a big favor. Here we observe Raskolnikov as the utilitarian, someone who believes that moral acts should be judged not in and of themselves, but in their consequences, on their social utility, and on whether or not they bring the greatest happiness to the greatest number.

It is this paradox of liberalism, with its belief that selfishness and extreme individualism are somehow socially beneficial, that Raskolnikov ultimately fails to reconcile. In his actions, he both reflects liberal thought and, at the same time, rebels against it. Raskolnikov's name is well chosen, derived as it is from the Russian word for "schismatic." He is torn between the competing impulses of urban society, between the need to assert one's individuality and his duty towards society as a whole. That he is unable to reconcile these impulses stands as a withering critique of liberal society that still has extraordinary resonance today.

One of the most important social criticisms was of the separation of the classes.   Dostoyevsky contended there were two classes--ordinary people and "supermen":

 Ordinary people must obey the law, but "supermen" — of whom there are very few in any generation — are entitled to break existing laws and make their own laws. Raskolnikov cites the French emperor Napoleon as the epitome of the superman type. He argues that Napoleon rose to power by overstepping the laws that govern ordinary people. Napoleon made his own laws and achieved his goals by killing tens of thousands of people in wars. Because Napoleon was a genius, Raskolnikov reasons, he was not regarded as a criminal. On the contrary, he was hailed as a hero. (Enotes)

Dostoyevsky felt that those with power could make their own laws and rules; however, the ordinary people were forced to obey these laws and rules, which was unfair. 

Another social criticism that some critics have brought up is Dostoyevsky's treatment of the "younger generation" in Russia.  Some critics feel he was degrading the younger generation and that they were destined for failure due to "their liberal ideas and natural sciences"  (Enotes).