The game is set in a parallel universe in the Devil May Cry series. It starts with Dante recovering from a hangover when he is attacked by a Hunter demon. Dante learns he is in Limbo City, a dreary city that appears calm at first, but transforms into a twisted parody of itself with demons and other monstrosities who frequently attempt to kill him whenever he is seen by the demons. After confronting a demon, Dante meets a girl named Kat who sends him to meet the leader of "The Order", an organization labeled by the press as terrorist. The Order is fighting against Limbo's demons and its leader Vergil expects Dante to join them. While fighting enemies Dante learns he is Nephilim the offspring of a demon and an angel. During the game, Dante is frequently taunted and stalked by the demon Mundus, who seeks to have Dante killed.
Source: Wikipedia, "DmC: Devil May Cry", available under the CC-BY-SA License.