The old pawnbroker is an evil person who is actually harming the poor people who come to her for pawning. According to Hegel, any harmful segment of society should be removed. Therefore, Raskolnikov reasons that by murdering the old pawnbroker, he will be removing a harmful "louse" from society.
If the ends are noble, the means can be justified. The old pawnbroker has a lot of money that will be "wasted" upon useless requiem services for her soul after her death. With that money, Raskolnikov will be able to complete his education and devote himself to the service of humanity, or he could distribute the money among needy and starving families, thus saving hundreds of people from ruin and destitution.
Dostoevsky had also apparently encountered other views of the Superman or Ubermensch — views that were not yet formulated in any coherent whole but were heard wherever intellectuals gathered. Svidrigailov was born from these ideas of self-gratification. Svidrigailov would reason: Since there is no will (or power) beyond that of my own, I must completely assert my own will until it is totally free of all restraint against it. Since there is no power beyond me that functions to punish, I am free to assert completely my own will.
Therefore, the Svidrigailov type of Ubermensch is one who possesses the strongest will and is able to make his desires and his power dominant over others. Svidrigailov can rape a 15-year-old girl and cause the death of a servant without any fear of punishment. He asserts his own will in order to gratify his own desires.
The test of this type of Ubermensch is that he must stand completely alone and must not allow his will to be influenced by the wishes of others. Thus, this assertion of the will isolates man from society. When Raskolnikov attempts to assert his will, he finds himself cut off from the rest of humanity. It is this dreadful solitude that Raskolnikov cannot stand and that makes him confess to become part of humanity again.
Raskolnikov's own theory adopts some of all of the above ideas with certain touches of his own. For Raskolnikov, all men are divided into two categories: ordinary and extraordinary. The ordinary man has to live in submission and has no right to transgress the law because he is ordinary. On the contrary, the extraordinary men have the right to commit any crime and to transgress the law in any way. They are extraordinary because they are the men who have the gift or talent to utter a New Word. It is the extraordinary men who forge civilization onward to new heights of achievements. The extraordinary man has this inner right to decide whether to overstep the law or any obstacle that stands in the way of the practical fulfillment of his idea, or New Words.
All great men capable of giving something new to society must not submit to the common law because if they do they cease to be great. Great men create new laws by their discoveries and therefore should have the right to eliminate a few men in order to make their new discoveries known to all of humanity. Thus, Raskolnikov "sanctions bloodshed in the name of conscience." (Raskolnikov constantly refers to Napoleon because Napoleon had the daring to commit various acts in order to complete his plans.)
Again it should be emphasized that, at the time of the murder, Raskolnikov had not worked these various theories into a consistent whole. All the individual parts were there, but some of the connecting details were missing. The murder was committed to see whether he dares commit a murder and therefore proves his will is strong.