Xbox One's upcoming backward compatibility will also extend to Xbox 360 DLC. Xbox Support tweeted out that the console will indeed be able to support ,but individual publishers will need to give permission first. It's also been confirmed by that cloud saves and Achievements will carry over, and 360 multiplayer will work as long as the servers are still running. What game are you looking forward to playing most on the Xbox One ?
@SomberShadowI Hey there, good question! While the console allows it, DLC of specific titles is up to the publisher :) ^GC
— Xbox Support (@XboxSupport3) June 22, 2015