The special attacks are done by holding the RT+RTstick in a circular motion. Depending on the level of special attack you have unlocked in the specials tab, the higher level the more damage, but also more bars of your Special bars consumed (green bars under your health). This can be filled by consuming energy drinks or by attacking enemies normally.
You can easily get this while playing through the story, but if your still missing some by the end go to the dojo select freestyle training and grind the rest out.
Note: most enemies won't die from a special attack alone so be sure to rough them up a bit first.