In order to turn into Hyper Shadow you first need to complete all the endings for all the stories. This includes the good and evil storylines. Once you do this, you will unlock the Last Story. Choose the Last Story and you will begin a level called The Last Way after some intermission sequences. Once you complete this and reach Black Doom, there will be an intermission sequence, at the end of which Shadow takes the Chaos Emeralds and turns into Hyper Shadow.

To defeat Devil Doom as Hyper Shadow, hold Square to charge Chaos Spear and hit it numerous times. Get lots of rings and you will survive.