Kassadin is what you want him to be.
For the most part, people tend to play him as an AP Nuke.
His Q and E skills have good AP-Magic Damage ratios, and can make for some very powerful burst damage. Combine that with a teleport that does AoE damage and stacks each time, you have a lot of damage potential. Most people will focus on building the Force Pulse, as it's an extremely powerful move. The weaknesses for this build is the fact that his cooldowns make his damage output negligible while they're on.
On the other hand, take a look at his W. Armor Pen Active, and Mana Regen passive. If you decide to work a full AP build, the mana regen is negligible when you compare them to the growing costs of your teleport, and the armor pen is near-useless since you do no damage when you build AP. His passive ability decreases magic damage by 10% and increases his attack speed depending on how much damage was converted.
And people use those abilities to play him as an AD character. His Q and E and R could be used for utility purposes rather than for damage. The silence ball and teleport means he can get the jump on a character while they can't do anything but auto-attack. Use his Armor Pen proc to increase the damage done with Kassadin's netherblade, and then use E so they can't run from him. And if they do, he can chase them with his Teleport all the while he wails on them.
Then there's the hybrid build.