DMC Dante usually only talked to himself during cinematics or while calling his attacks though.
There are certain games where I cherish the protagonist's commentary they make to themselves as an important part of what defines the overall experience. Games like Duke Nukem 3D, Blood or Max Payne wouldn't be themselves without the protagonist making wry observations and corny jokes now and then, because it helps to illustrate the kind of character they are.
Then there are games where the protagonist talking to themselves can have the opposite effect. Sometimes the emotions and perspectives of the protagonist can be told far more aptly by their body language, or conveyed through gameplay mechanics. I maintain that one of the New Tomb Raider's problems is that Lara vocalizes more than is perhaps necessary to demonstrate the feelings she is going through or to prompt the player towards the next objective.
DmC is intended as more of a character piece than it's predecessors, but I seldom get the feeling that Dante's near-constant monologuing really illustrates biting insights on his perspective we couldn't get through gameplay. To me it feels more like the game is filling the dead air with chatter in a desperate attempt to keep our attention, as if we'd lose interest in the game if the protagonist wasn't busy flapping his gums every other second. You see this sort of use in voice work in a lot of children's cartoons.
It's sad, because the game clearly has a lot of care and attention paid to it's environments, and being allowed to soak up the stark desperate atmosphere of being alone in a hostile environment would be better complemented if Dante kept his mouth shut.