I think this is an example of a relatively small genre of movies, which I'm classifying as "undomesticated male gets civilized/redeemed by cute kid." Let's see if I can find that at TVtropes.org... Well, this is close: "Children Raise You". But that's not on the tropes page for the movie itself. Hm.
Anyway: Bill Murray plays Vincent, who's a mess: a grumpy misanthrope who lies, steals, drinks too much, smokes too much, frequents a Russian hooker (Naomi Watts, with heart of gold) and plays the horses poorly. His Brooklyn accent comes and goes, but that's OK. One day, he acquires next-door neighbors: Maggie (Melissa McCarthy) and her young son Oliver (Jaeden Lieberher), who are in flight from Maggie's cheating husband.
Maggie is also on shaky finances, and has a demanding job that keeps her away from home a lot. Oliver is kind of small, a ripe target for the bullies at his Catholic school. They relieve him of his keys, wallet, and street clothes, leaving him no alternative than to throw himself on Vincent's mercy. And they do the sort of thing that happens in this kind of movie: learn from each other and improve each other. There are dramatic overtones throughout, but it's mostly funny.
I can't imagine anyone doing better in this role than Bill Murray. And Melissa McCarthy does a fine job too. (I was surprised, because I've only seen her in roles where she's loud and rude. She can do other things!)
Everybody else is good too. Special mention to Chris O'Dowd who plays Brother Gregory, a humane and funny teacher at Oliver's school.