Okay, first of all, I suggest you go and visit this thread, by
@DragonMaster2010:
http://devilmaycry.org/threads/the-character-range-of-dante-from-kamiya-to-tameem.19787/
It should be right what you need.
To the matter at hand, I'll start by quoting
@berto from that very thread (check out his posts btw, he does a good job at expressing his opinions).
I agree with most that's said there, but I'll add something more.
At the beginning part of the game, Dante's presented as a superpowered badass that enjoys kicking demons' butts, and that's all it is for him: fun. He's someone who lives for it (duh, eh?) and is always ready and willing to pick up a challenge. As you said, doesn't care about much else. A challenge from his brother? Hell yeah he'll pick the gauntlet.
So there his journey begins, and everything's a cakewalk (for him, not for the player LOL), it's a party, he goes at it so lighheartedly, so uncaring, nothing's on his mind.
On the way, he meets Lady and in his uncaring and lighthearted fashion, cracks jokes trying to hit on her, even kind of ridiculing her.
"Let you go?! But it would be a waste if you ended up as just a pretty stain."
As I said, who cares, it's all fun.
At the top of the tower, Vergil awaits him, and here is one of the turning point for the story and Dante's character.
What's his attitude towards Vergil in this? Same as always, he cares little about him, is a showoff, playful, mocking, light. Doesn't take him seriously, he's very confident, arrogant, actually.
When he's questioned about the acceptance of his heritage and his father's legacy, he straight up refuses to even acknowledge he has a father. Means nothing to him.
After his defeat, the change starts to boot up. Allow me a computer term. I do study Computer Science after all. Now he's got an actual drive, as selfish as it is: to find Vergil and get his payback.
He meets with Lady again, and once again he's joking and acting uncaring, but when they stumble upon Arkham's body they confront for the first time. They fight and Lady lets her anger out, making Dante know about her problems with her father and the kind of horrible man he was, expressing the deep hatred she feels towards him, without failing to shove Dante's demonic heritage in his face and questioning his understanding of the value of family becasue of it. By the end of it, we're left with a pondering Dante, who's now
beginning to be bothered by something else than his own problems.
"Family huh?"
The time for payback comes, and here we can notice how Dante takes the situation more seriously than the first time. He still cracks a couple of jokes, but it ain't a crazy party anymore and he's more determined this time. He also stops to refuse to acknowledge his father. He makes fun of him by calling him "Pop", but doesn't act as if he never existed anymore.
He's not screwing around now, this battle is not for fun like the first time, and he shows this by harshly telling Lady to begone. The party boy Dante from the beginning wouldn't have cared, would have been like "the more the merrier", but this ain't the case.
After Jester revealed his true self and Dante got beaten and found himself outsmarted by Arkham by being one of his pawns, he saves Lady's life and he questions once again the value of family as he witnesses Lady's determination to take down her father to avenge her mother AND stop the apocalypse he's trying to unleash.
"
- Are you gonna go?
- Yes. I'm going to finish him off.
- Well, you might as well forget it, cause you're no match for him.
- Regardless, I must go. I had a chance to stop him before, but I couldn't. I'm responsible for all this mess.
- Responsible! Does that bother you that much?
- He's my father! Besides, who else can undo what he's done? ... A demon like you wouldn't understand.
- Family huh...
"
The seed of awareness of family's value and sense of duty that she planted in him before is beginning to grow.
Their next encounter is decisive. Dante's not the self-absorbed guy he was initially, he doesn't only care about himself anymore and confronts Lady's stubborness to try and dissuade her to basically commit suicide.
"I told you, you can't do it! Don't you get it? This is not a human's job."
It's after the fight that it's more evident than ever that he's now abandoned his ego-fullfilling attitude and that he now does what he does cause of his own sense of duty and because he feels he has to fix the mess Vergil started. They both soften up and after the fight that was basically an outburst of emotions (Lady's stubborness and proud, and Dante's determination to pursue his quest).
"This whole business started with my father sealing the entrance between the two worlds. And now my brother's trying to break that spell and turn everything into demonville. This is my family matter too."
He feels it's his responsibility, he's acknowledged his family's heritage, come to terms with it, and accepted the burden, by witnessing how Lady had relentlessly carried her own family related burden.
"Quite frankly, at first I didn't give a damn. But because of you, I know what's important now. I know what I need to do."
Accepting Lady's request and to take her burden too is like sealing the deal of his commitment to the cause. He's grown to care about her and to sympathize with her problems too.
"
- Dante... please, free my father.
- I will... Lady.
"
Now to the final battle. You don't need me to tell you that this is arguably the most memorable part of the franchise for the vast majority of fans, also because of its significance.
Remember Dante's attitude in this same kind of situations the first time? Well, compare it to his now.
He now tries to get Vergil to reason, he doesn't wanna fight him unless it's the only option, unlike the previous two times, where he was all for a fight. Unlike the previous two times, he now fully embraces his father's blood legacy (which is also symbolized by picking up his father's sword), unlike the previous two times, he doesn't mock it, and he's fighting for a cause. And not a selfish cause, like fun seeking or payback.
His attitude towards the situation is different, his attitude towards the ideals of family and duty is different, and his attitude towards Vergil himself as an adversary and a brother is different. This Dante is one who reaches his hand out for his falling brother, who is sorry for his loss and cries for it despite all the evil Vergil did.
Notice how the camera doesn't frame his eyes by the way, there's no eccessive display of sorrow, it's subtle, which makes it all the more effective.
It all began with "Let's go to a crazy party, I'm sure you've got something fun for me, eh Vergil?" and ended with a Dante who is more aware of what's important, who accepts his heritage, more caring, with more empathy, who takes stuff more seriously, a Dante who conserves the cut and blooded gauntlet he tossed at Vergil in response to his, in his memory.
A prototype of what DMC1 Dante will be years later, and more similar to him than he was at the beginning of the game. He's now got the attire he sports in DMC1, we get why he now uses Force Edge, and where the shop's name comes from, what it means.
As to why DmC Dante and DmC in general are so hated, I think it doesn't stem from simply the fact that they rebooted the franchise. No, it's my belief that it comes from the fact that they did it without giving DMC proper closure. They introduced a new character in DMC4 and DMC2 ended with a cliffhanger, fans wanted more, and Capcom responded with "Reboot!"
Ask yourself this: do you think people would have gotten so mad if the main series was given proper closure? I think not. Sure, there would have still been a group of disgruntled fans moaning, but I'm sure it wouldn't have been as bad as it turned out.
Yes, some traits of DmC Dante's personality do contribute to people not standing him, like swearing and booze, and "thug life" kind of attitude even in the later parts of the game, which are generally considered cheap ways to make a character cool, but I do think the main problem is the one I said above.