With the exception of Devil May Cry 2, Dante is very flippant, frequently considered a show off, as seen performing exagerated moves and trash talking his opponents regardless of their strength.
Dante is fairly laid-back, as seen in the anime he typically refuses any job he dislikes, regardless of how much it pays, but will immediately take on any paranormal cases, especially those involving demon hunting, with or without pay. He has a strong sense of justice and can be counted on to do the right thing, despite his complaints.
Dante loves pizza, there's always one in his office, his liking for pizza is frequently used as joke through the games and other media. He also likes strawberry sundaes, the anime points out that these two foods are basically all he eats. Dante has the habit of getting drunk, mentioning in the manga that he got to the point of proposing to a mop once.
Music seems to be one of his hobbies, judging by his demonstration with the Devil Arm guitar Nevan. He also has a drum set and a guitar in his office.
As for in DmC, Dante has a similar personality to the original, but as a younger version with some changes in his past, the new Dante is fairly more rebellious, treating people with a more spiteful tone, but still showing a caring side for his associates. There's pizza in his trailer during the intro cutscene and a Combichrist poster, showing his interest in pizza and music. There's reference to his drinking habits in the intro sequence as well.