It sounds like it's not booting correctly... I would try these steps but if the don't work the system may just be "bricked"
Step one: inshore that the hdmi cable is not faulty/plugged in correctly with no problems (if the Xbox does not detect a output source it will not boot)
Step two: try to reinstall the os, you can do this with a USB by going to...
...and following these steps...
Copy to a USB flash drive
If you can't connect your Xbox 360 console to the Internet, you can copy the console update from your computer to a USB flash drive and then install it on the console.
Note The USB flash drive must use the FAT32 file system. For information about how to format a USB flash drive, see the “Format a USB flash drive” section of Xbox 360 USB flash drive support.
Plug your USB flash drive into a USB port on your computer.
Download the update file.
Click Save to save the console update .zip file on your computer.
Unzip the file. If you're using Windows XP, Windows Vista, or Windows 7, double-click the .zip file to unzip it.
Copy the contents of the .zip file to the root directory of your USB flash drive (for example, E:\). Do not copy the contents to another folder on the USB flash drive (for example, E:\Files).
Unplug the USB flash drive from your computer.
Plug the flash drive into a USB port on your Xbox 360 console.
Restart the console.
When the console restarts, the installation program starts automatically. When prompted, select Yes to update the new console software.
If that doesn't work try it again without the HDD connected
And if that doesn't work I am afraid that the console is most likely "bricked" and you will need to sent the console to Microsoft for repairs.