So first of all: the circumstances that Dostoyevsky puts Raskolnikov (the protagonist) in:
Raskolnikov lives in a poverty-striken garret. He only moved there to have the money to go to university and get a good-paying job to help his family back at home--his mother and sister, with no mention of his father. They looked at him as their saviour, too. He was smart and handsome and had great potential, hence why they sacrificed themselves to send him to university.
But because of his wretched state, arrogance and pride, he was too ashamed to be seen in his rags, he did not socialise with the "peasants" at his university, and refused food from his landlady, letting himself starve. Clearly, this only makes him spiral downhill.
Second, the pawnbroker he meets:
2.1: His erroneous view of the pawnbroker's sister as a slave.
This pawnbroker lives with her disabled sister who, as Raskolnikov erroneously sees it, is a slave to the pawnbroker.
He thinks this because the disabled sister cleans the house and organises everything for her older, rich sister--as if she's a full-time maid/slave. But what Raskolnikov doesn't realise, which we as the readers can, is that her disabled sister has no way of getting work in this time and place given her condition. Additionally, the small flat in which the pawnbroker lives has two equally-sized rooms, and the disabled sister gets one all to herself, while the pawnbroker has her room filled with the jewellery and such that she traded, so much so she has to fill the underside of her bed with it.
2.2: His erroneous view of the pawnbroker as a gold-hoarding predator
Raskolnikov's view of the pawnbroker is no different from a dragon hoarding gold and enslaving a beautiful young princess (the disabled sister).
One reason why he thinks this is when he meets a drunk, Marmeladov, who is in pure suffering living with his wife and her three small children. This wife had something going for her when she was younger but lost it all. To keep a memory of who she was and what she earned, she kept a gold medal by her side. But eventually, they had to sell that (obviously, to a pawnbroker).
And so, Raskolnikov sees the pawnbroker as preying on the neediest, and only perpetuating suffering. It's as if the people only went to her to regrettably sell their precious goods out of necessity, and she gobbled it right up.
But Dostoyevsky describes a circumstance that the reader can infer to realise Raskolnikov's many irrationalities. E.g.:
‘You come with such trifles, my good sir, it’s scarcely
worth anything. I gave you two roubles last time for your
ring and one could buy it quite new at a jeweler’s for a
rouble and a half.’
Raskolnikov also ignores the fact that this old woman doesn't gain anything from having so much jewellery lying around. E.g., when he robs her:
He left the chest of drawers, and at once felt under the bedstead, knowing that old women usually keep boxes under their beds. And so it was; there was a good-sized box under the bed, at least a yard in length, with an arched lid covered with red leather and studded with steel nails.
[...]
But no sooner did he touch the clothes than a gold watch slipped from under the fur coat. He made haste to turn them all over. There turned out to be various articles made of gold among the clothes—probably all pledges, unredeemed or waiting to be redeemed—bracelets, chains, ear-rings, pins and such things. Some were in cases, others simply wrapped in newspaper, carefully and exactly folded, and tied round with tape. Without any delay, he began filling up the pockets of his trousers and overcoat without examining or undoing the parcels and cases; but he had not time to take many….
And lastly, so much for "saving" the princess from the "dragon" that enslaved her:
In the middle of the room stood Lizaveta [the disabled sister] with a big bundle in her arms. She was gazing in stupefaction at her murdered sister [the pawnbroker, of course], white as a sheet and seeming not to have the strength to cry out. Seeing him run out of the bedroom, she began faintly quivering all over, like a leaf, a shudder ran down her face; she lifted her hand, opened her mouth, but still did not scream. She began slowly backing away from him into the corner, staring intently, persistently at him, but still uttered no sound, as though she could not get breath to scream. He rushed at her with the axe; her mouth twitched piteously, as one sees babies’ mouths, when they begin to be frightened, stare intently at what frightens them and are on the point of screaming. And this hapless Lizaveta was so simple and had been so thoroughly crushed and scared that she did not even raise a hand to guard her face, though that was the most necessary and natural action at the moment, for the axe was raised over her face. She only put up her empty left hand, but not to her face, slowly holding it out before her as though motioning him away. The axe fell with the sharp edge just on the skull and split at one blow all the top of the head. She fell heavily at once. Raskolnikov completely lost his head, snatching up her bundle, dropped it again and ran into the entry.
The only predator, in the end, was Raskolnikov.
Whether or not Raskolnikov really killed Lizaveta there and then was not as important--she was dead anyway without her older sister to provide for her. Raskolnikov himself had talked to people on the verge of death in that town from being unable to care for themselves. We have no reason to think that this incompetent, disabled woman would last long by herself.
Third of all, the circumstances Raskolnikov's family was in.
I could also post another monologue by Raskolnikov in his own head after reading that his sister was driven to a circumstance where she convinced herself it was the right thing to do to marry an old rich man who can both provide for her poor family and also hire Raskolnikov (her older brother who she somewhat lives for)
So he thinks that the perfect plan that'd better his wretched state, stop his sister from prostituting herself and relieving his mother, as well as help the people in the city form being explotied, and of course, free the princess from teh grasp of this dragon, he plans to kill the old pawn broker.
What's amazing is that from Raskolnikov's POV, things appear perfectly logical. But Dostoyevsky's more general and impartial description of the situation lets us see (if we pay close attention) the very typical human flaws in his views.
And when he sees that she has almost no space left in her one, small room because of these jewellery... the thought doesn't cross his mind that (1) she lives the entire other room to her sister and (2) has some issues herself, as she's obviously not profiting a lot from this pawnbroker business and
And why does she write to me, ‘love Dounia, Rodya, and she loves you more than herself’? Has she a secret conscience-prick at sacrificing her daughter to her son? ‘You are our one comfort, you are everything to us.’ Oh, mother!’His bitterness grew more and more intense, and if he had happened to meet Mr. Luzhin at the moment, he might have murdered him.