I do believe this feature is unlocked after the first, or maybe the third wish. Next to your readout at the top left is a large circle. From bashing the brains out of Aberrations, you will start to fill this gauge up. Once it is topped out, it will flash and you will have the "God Mode" option in your menu to change into. To get the trophy, all you have to do is use it and bam, you'll become a familiar form.
NOTE: God Mode is rather overpowered and is duly made that way with flaws. You will be fully healed of any ailments, full HP/SP restoration, and your power will increase. All of your Divine Skills will be replaced with an attack that hits the entire room for quite a lot of damage. However, this mode will drain your Endurance much faster, and should you use the Godly Divine Skill, you will revert right back into your normal form. Only use this as a last-ditch recovery to clear a room, as if you stay in it too long, you will be worse off if you still have more floors to go down. (And it doesn't seem to be too helpful against bosses, either, but that may just be a bad experience I had and not reflect the bosses in general.)