People often forget that Dante was both orphaned and institutionalized in DmC, meaning he probably encountered other delinquents, not just demons. He also pretty much raised himself without any real authority/parental figure, meaning he never really matured. These elements actually serve as a logical basis for his immature behavior and impulsive nature in the beginning of the game. That's why he swears like a drunken sailor and maintains a prickly, pretentious, arrogant, self-serving attitude in the beginning of the game.
And you want to know the best part? Other characters are aware of his behavior...and they regard it with a level of distaste. When Dante brashly tells Kat that he "didn't ask for her help" at the end of Mission 1, you can see her briefly roll her eyes. When Dante promptly states that he "doesn't give a ****" about Vergil's efforts to free humanity, so can see Vergil bite his lip like he's really trying retain patience and tolerance. Kat even comments at the beginning of Mission 2: "Are you sure this is going to work? He doesn't seem to care."
Dante's supposed to be the somewhat-resentful street urchin in the beginning of the game, because that's the person he's been up till this point. He doesn't have a reason or an obligation to care about anyone else...and only seems interested in partying and scoring girls. (I can't help but wonder if Ninja Theory was making subtle commentary of the Old Dante's lifestyle of "crazy partying" at the beginning of DMC3). But you can tell in the cutscene where the Hunter mentions his mother in Mission 1, Dante suddenly pauses in the midst of his vulgar, fun-loving spree and remembers his mother. He suddenly looks wounded...insecure...confused.
And that's when the changes start happening.
Gradually, over the course of the game, Dante starts treating his comrades with mutual respect, even cracking the occasional joke with them. He starts swearing less and less over the course of the game...and shows his willingness to help and his tolerance towards others when he helps Phineas, despite his original mistrust of the demon. He stops being self-centered and spends most of the game asking his comrades about themselves....their lives, their histories, what makes them tick.
But the most predominant evidence of change comes in the form of Dante's attachment to Kat. He doesn't hit on her or use the same sleaze he'd employ when scoring girls at the local club...he befriends her. He empathizes with her situation, asks more about her life before the Order, comes to understand the burden of her medium abilities. She becomes the sole factor behind his drive to fight the demons later in the game, as hinted by his fierce reluctance to abandon her in Mundus' hands like Vergil wants to. His desire to rescue her brings out some better qualities in him....like when he dismisses the stripper in Lilith's club, or when he decides to spare Lilith and use her for a fair exchange. And by the time Mundus is defeated, and Vergil's true intentions come together, Dante shows his newfound selflessness by trying to bestow Kat with some credit for the world's salvation.
This change in Dante's character over the course of the game was something I always wanted the Old Dante to have, but he never really got. The closest it came to his character changing was in DMC3, but all the plot did was act as a foundation for Dante's demon hunting business. By the end of the game, he didn't show any real changes...and he just seemed like the same arrogant, over-the-top manchild he was in the beginning of the game. Sure, Dante got a more professional tone in games like DMC1 and DMC4, but that was more because of the series being handed to multiple teams with different writers, rather than it being a form of overall development for the character.
DmC's Dante got the growth, change, and evolution that I had always wanted. By the end of the game, he discards his old attitude and asserts himself as the protector of the human race. He becomes the Demon Hunter we all wanted him to become, and it didn't take a single flashily-choreographed cutscene or corny one-liner like "Devils Never Cry" to do it.