How do you get into the underworld on mythology island?

The entrance to the Underworld (realm of Hades) is in the Temple of Hades. Put pomegranates (from the garden of the Sphinx) on the altar to open the door.

The only path to the River Styx is a vertical fall into the bottomless pit. Cross the river and get a whisker from Cerberus by playing the reed pipe (blue-green-yellow-blue-red-green). You cannot enter the Hades throne room until later when Hercules is helping you.