Some of it comes from Resident Evil. Funny as it may sound the first Devil May Cry was a failed attempt to be Resident Evil 4. But they finally decided that this game shouldn't be a Resident Evil title, but since it sounded pretty good they changed it even more and Devil May Cry was born.
There are still similarities between Devil May Cry and Resident Evil 4, and plus, most of Devil May Cry 4's Easter Eggs are from Resident Evil 4.
The storyline of the original Devil May Cry was adopted from Divine Comedy. The similarities are great. Even in names. In Divine Comedy Dante, the main character is invited to Hell by Beatrice and is guided through it by Virgil. I assume that if you've played Devil May Cry & Devil May Cry 3 you've already noticed the similarities.
Dante's character in the original Devil May Cry was influenced by another manga character. The manga character was an investigator of weird cases who often spoke cocky/funny lines while spotting interesting/weird things or while speaking with adversaries. Almost like Dante in the game who was a nice quote for every book/statue/item and of course every boss in the first game.
The Devils where created as something different. Since this wasn't gonna be a Resident Evil game, it couldn't have Zombies/possessed people/humanoids altered by viruses and the rest. So they adopted something else. And Devils came into the game. They tried to find appropriate names/appearances/moves for every enemy in the game. For example, the Beelzebubs (I may have wrote it wrong) are different kinds of demonic flies in the game. But if you search for Beelzebub he was apparently some high ranking devil and the lord of flies. So you get the idea.
Devil May Cry universe is in a place a bit different from our own. There are signs that they do have advanced technology through Dante's guns or through cars/airplanes and the rest spotted throughout the different Devil May Crys. However, there are certain religious thingies about devils scattered all around the games. Also even though there are modern houses, castles and places that belong to different ages exist in the Devil May Cry universe as good as new.
Nothing else comes to my mind right now, I hope I helped a bit.