Damn I get the feeling like I answer this every five minutes.
First of all. To do the Paradox Cups. All you have to do is enter the Cave of the dead and go all the way to the end to Hades' Chamber. Talk to Hades and he will host the Paradox Cups.
Pain and Panic Paradox Cup: Default after you beat every regular tournament. (Goddess of Fate, Titan, Cerberus, and Pain and Panic.)
Cerberus Paradox Cup: Have all forms at level five.
Titan Paradox Cup: Have summon at level five
Hades Paradox Cup: Be at Level 99; Have all Forms and Summons maxed out. And all other tournaments must be beaten.