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Marmeladov is the kind of character that appears most often in after-school specials. He's a cautionary tale—take one sip of beer before your 21st birthday, and you'll end up just like Marmeladov. (Brr. We're scared straight now.)
Dude has a problem with the sauce, to put it extremely mildly. His drinking strongly contributes to huge suffering...for his entire family. His daughter Sonia has been forced into prostitution, and he uses her hard-earned money to buy booze. He claims to love his family but can't get his drinking habit under control.
It's a little surprising that Raskolnikov seems to care about him so much and that he treats him so gently. Perhaps it's because his suffering and helplessness is so apparent—as we know, Raskolnikov has a soft spot for stuff like that.
In any case, if Raskolnikov hadn't helped Marmeladov, he might never have met Sonia. Also, Marmeladov brings out something in Raskolnikov that we don't often see on display in our axe-murdering protagonist: real tenderness and compassion. Raskolnikov actually acts on those feelings in a positive way and tries to help Marmeladov.
When Raskolnikov leaves Marmeladov's house after Marmeladov's death, he meets Nikodim Fomitch, who notices that Raskolnikov is "spattered with blood." Raskolnikov says, "with a peculiar air, 'Yes...I'm covered with blood'" (2.7.92).
(To be fair, Raskolnikov is covered with blood more than the average butcher.)
This creates a sense of doubling—or even déjà vu—and recalls the murders. This is the kind of thing that makes Raskolnikov so confused. On the one hand, he sees that there's a huge difference in getting blood on your clothes from killing someone and getting blood on your clothes from trying to save someone. Yet, his mind can't quite grasp why there's such a difference.
From a character analysis standpoint, Marmeladov is a bit of a mystery wrapped in an enigma wrapped in a beer coozy. If it wasn't for the fact that Raskolnikov liked him so much, we'd be tempted to call Marmeladov a bad guy. But, as much as the novel disapproves of alcohol (just say no, y'all), it seems to consider alcoholism a genuine illness and, as such, places Marmeladov in opposition to people like Svidrigaïlov and Luzhin...who appear to get off on being mean and abusing their power.
But Marmeladov? Marmeladov just gets off on downing the hard stuff like there's no tomorrow.