If I cast a spell without paying it's mana cost, can I cast it for alternate or additional costs? This is for a Vadrok deck if the manner in which the spell is cast matters. I just want to overload Cyclonic rift from my graveyard every turn.
When you cast a spell without paying its mana cost, you're already paying an alternate cost of 0. Any time you're casting a spell and not paying the printed cost, that's an alternate cost and you can only ever pay one alternate cost to cast a spell.
So why can't you choose to overload Cyclonic Rift rather than casting it for free with Vadrok? That's because Vadrok's ability is what's allowing you to cast the spell at all, so you have to pay the alternate cost it dictates if you want to cast it.
Additional costs such as Kicker costs are different, however, since you can always choose to pay an additional cost for a spell no matter what cost or alternate cost you're using to cast it in the first place. You still need to pay the mana for these costs, however. For example, if you cast Jilt off of Vadrok's trigger, you can pay 1R to get the kicker effect, but you can't get it for free.