Well with me, I'd always assume that the reason Dante's so different in each game is more on time, and events.
DMC3-He's gotten too big for his boots, and believes he's the baddest thing on Earth, but after going through temi ni gru, having to see Lady's struggles, and being forced to possibly kill his own brother, could have made him a bit more serious.
DMC1-Now he's well rounded and is use to the demon hunting while still having a little fun with it. However that all changed when Trish came in and he felt a deeper feeling inside him that's made him a bit more solemn, but at least relieved that Trish was okay. Though the fact that he had to defeat and possibly kill his brother again hit him hard. And then Trish left which made him feel a bit lonely and bored as well, but also sad too.
DMC anime-We see Dante now bored with hunting since there's no challenge for him to take. He takes demons down with no issues while still being bored in his office. But a warm sunshine called Patty comes in to try and open him up. And with every episode, we see Dante smile more and more. With Brad and Angela, he felt a remembrance of his mother and father because Brad was a demon who was in love with a the human woman Angela. And Trish started showing up and Dante felt a bit more happy again. And finally at the end when he's more then psyched for another mission is him being more ready to go.
DMC4-Though I find this Dante the f*ck up of all the Dante's, I think I have a reason why he is how he is. At first, Dante wasn't to interested in doing the mission when Lady presented it. Plus he seemed more content now that Trish was back with him. But when Lady mentioned them worshiping a demon as a god (Sparda) this peeked Dante's interest for a second, and finally decided to get it over with to keep him occupied. But then Trish took the Sparda, and he was forced to go. Getting there, he finds out that there's another person who's related to Sparda, which gets him happy that he's found Nero, but decides to go back to his antics from DMC3, because he figured he'd be cheeky with Nero as he was with his last blood relation to his father Vergil. Now, I thought it was dumb that he didn't even worry or lift a finger to help Nero when he was getting absorbed by the Savior, but hey, that's just how writing can be in the eyes of Capcom. Any who, Dante finally got Vergil's sword to Nero but when Nero didn't respond right away, Dante got worried and when he finally felt Nero again, he urged Nero to end this so that he can destroy the savior and make sure Nero still lives. At the end, Dante was content and left in a better mood knowing that he wasn't alone in the world.
DMC2-Now for DMC2, I believe that it has something to do with everyone. Lady probably passed away from one of her missions. Humans get old, and sometimes they can't do what they use to back then and thus she slipped up and was killed by a demon whom Dante must have massacred. Then Nero was next to suffer as he must have almost died but instead lost his demon arm or it was cut off of him, thus breaking his ability to use Yamato and it then felt as though Dante lost Vergil all over again. And finally Trish dying or sacrificing herself for Dante, which hits him again because he loved Trish like a lover and as a mother, and him seeing Eva die all over again put him in complete depression. Thus leading to years in the future, where Dante is alone and no one visits him. Morrison probably died of old age, and Patty probably moved away. So when he heard of a mission in Vidu Marie, he thought he might as well accept. He still has a job to do.
But that's my thoughts on it.