Microsoft has confirmed that for all of the games that are backward compatible, their respective DLC will carry over and work on Xbox One.
However, Xbox 360 DLC will not be purchasable on Xbox One. A Microsoft representative confirmed that you'll only be able to buy Xbox 360 DLC on an Xbox 360 or through Xbox.com
"Right now, Backward Compatibility does not emulate the Xbox 360 store and new Xbox 360 games can be purchased on your Xbox 360 or via Xbox.com," a company rep told Game Informer.
"We will support both digital and disc-based entitlements for Xbox 360 games running on Xbox One," Microsoft told the site, who also confirmed in their report that 360 DLC will automatically download to your console along with the games.
Xbox One's backward compatibility launches today, a few hours after the new dashboard update. As Major Nelson notes, here's how to get it:
If you use the Instant-on power mode (available via Settings - Power & startup), your Xbox One will take the update in the background when you put your console into Connected Standby mode. As this is a global rolling launch, consoles in Connected Standby will be updated at various times throughout the day based on your time zone.
If your console is in Energy-saving mode, your Xbox One will not automatically download and install the update. To initiate the download and update process, you can go to Settings - System to manually check if an update is available and choose "Update Now".
Note that all Xbox One consoles will receive a prompt for a mandatory update on November 23rd if they have not yet updated to the New Xbox One Experience.