Instructions
Turn on your Xbox 360.
Navigate to Microsoft's License Migration website. Click "Start The License Transfer Process Now." Enter your username and password. Click "Next."
Sign in to Xbox Live on your Xbox 360 with the gamer profile you have used to purchase the content. Your Xbox 360, in conjunction with your PC, will begin updating your DLC licenses. The Xbox 360 will prompt you when it has finished and this process takes 30 minutes. Click "Next" on your computer when finished.
Click "Confirm." This screen is just to confirm that the process worked and that this is the console you want your new licenses associated on.
Press the "Guide" button on your Xbox 360 controller.
Select "Settings" and "Account Management." Select "Download History" in the list provided.
Select each piece of DLC that you've bought. Select "Download." This will re-download each piece of DLC onto your new system with the updated license, allowing you to play all of your DLC on your new Xbox 360.