Yes, the Xbox One backward compatibility feature supports DLC.
Taken from Major Nelson's blog:
Q: Will the DLC for backward compatible titles be available too?
A: Yes. If you own the DLC, you can install and use it just like you do on the Xbox 360. To install DLC you own, go to the Manage Game section for that specific title and install the DLC to your drive.
From personal experience (Fallout 3 GOTY), the on disc DLC was also installed. I believe I needed to put in the 2nd disc to get the DLC to install.
As for digital copies, I would suspect you get the DLC if the digital copy includes the DLC.
