According to a tweet made by Xbox 360 Japan, the first DLC for Resident Evil 6 will land on the Xbox 360 first. For now there is no mention of when PS3 will get their DLC. According to speculations the PS3 player base will have to wait out at least two weeks before the DLC lands on their console.
Here is a rough translation of the tweet (below):
360 version of it in “Resident Evil 6” Xbox is also planned with the prior delivery of downloadable content. I do not miss the information in the future!