You will eventually get a spell or two through story progression, but you can also purchase the spells with your souls. I was playing as a Void member so I had two spells at one point that I could alternate between. This popped for me after only a few hours.
For my Void Caos I definitely needed to use the spells to get past tougher enemies, but I often found myself not using them to actually perform the killing blow. Just make sure you use your spells when they are recharged and make sure it lands the killing blow.
Update October 3, 2013
According to Rourc if you kill an enemy that causes you to level up this also seems to count as killing an enemy with a spell.