If you own DLC for any of your Xbox 360 games, it will still work on Xbox One. If you don't, you won't be able to buy it from the new console.
Microsoft confirmed before that all the DLC you bought for your Xbox 360 games will carry over to Xbox One, if the game is backwards compatible.
In cases where you don't yet own DLC for a game, you won't be able to buy it through the Xbox One console. A Microsoft representative confirmed to GameInformer that you can only buy said DLC from 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," the representative said.
If you want to navigate to the DLC you already own on Xbox One, you'll find it under the 'Ready to Install' section for each game. You can get there by clicking 'Manage Game'.
Xbox One's backwards compatibility officially launches today, 12 hours after the new dashboard becomes available.