The Chosen One
Well the short version, is you cant start one unless you are subscribed because you have to travel to other cities, and only subscribers can. After that part, you have to get to a minimum of 100,000 xp points and be able to do AI level cases. Then you'll have to seek out the ambassador and get an invitation to Shangri La. From there you will do something in Shangri La and get to start a villian hunt.
Villain hunts are done solely by yourself and you are offered no help from any other play or people in your agencies. The only help you will essentially get is from your contacts (which by the way is recommened you have all your contacts before you go on a villian hunt).
They can also be very costly, so it's always best to stock up on cash because you will spend a lot in travel expenses and bribes.