What you have to do to download dlc dragons teeth:
1. Be premium member
2. you must be updated with current patch for xbox one (It should show the Chain Link game mode under quick match)
3. If you are updated with the current patch attempt to start a game under the quick match in the Chain Link game type
4. You should then be brought to a screen that tells you that it isnt currently installed but is available to download select and go to download
5. After you select this IT WILL bring you up to the same area where you would normally purchase it ......BUT since you are premium it will say buy for $0.00 <then install (it should not tell you it costs to install because atm it is premium only which it is free to and is yet to be released to rest of community for another 2 weeks, if it says other than that you should probably check with dice/ea etc about your premium membership)
6. Game should install fine unless you are downloading other items
7. Then make sure you have quit the current battlefield game you are currently in after install is complete (I only say this because the way xbox one works is it keeps you ready to play at all times so it really never restarts the game but resumes it as if it never shut off sorta like when you close a laptop where it goes into a low power state yet you can open it and resume) easiest way is to go to game push the button with 3 bars on it(old start button) and select quit
8. after you quit the game go ahead and restart and you should be ready to play
This should work for all it is pretty in depth in case some people dont know exactly what items i was talking about. I have done this method for all of my installs for DLC and have had no issues with it what so ever. Hope this helps everyone out! :D
This may also work for playstation though i cannot be for certain though you may try similar steps.
Let me know with XP if this works it worked for me on xbox one and xbox 360