Kassadin is the most badass champion in LoL, now and forever. He is immensely strong, has Flash as an ulti with a ridiculously low cooldown, and his passive shuts down mages like nobody's business. Now that his coolness is established, let's break him down:

Weapon Mastery (Innate) absorbs 15% of magic damage and converts it to attack speed. This gives Kassadin the immediate upper hand against other AP champs.

Null Sphere (Q) deals flat magic damage (+0.7 AP) to a target and silences them (starting at 1 second). A strong ranged attack that turns off abilities... A very powerful ability that is useful from start to finish. The first part of his deadly combo.

Nether Blade (W) passively restores mana for every normal attack, and when activated gives Kass an armor pen. boost for a period of time. This is the last thing you level, its usefulness pales in comparison to his other abilites.

Force Pulse (E) passively gains charges each time a spell is used near Kassadin (including his own spells). Once he has 6 charges he can use this to deal damage to all enemies in a cone for flat magic damage (+0.8 AP) and slow them by a significant percentage (starting at 30%). The second part of the lethal combo with which Kassadin racks up kills in many games, it deals heavy damage and keeps enemies within striking distance.

Rift Walk (R) is flash, except it deals damage upon arrival and starts with a 6 second cooldown (reduced by 1 second per level). The final piece of the combo, the escape mechanism, the initiator. This makes Kassadin, well, Kassadin.

The combo which I have referred to is one of those delightful 1-2-3 combos (Q, R, E).
First, you hit them with the Null Sphere, silencing them from countering. Next, you Rift Walk right next to them, dealing additional damage on arrival. Using these adds charges for your final attack with Force Pulse, which will either kill them or leave them at extremely low health. If it doesn't kill them, then simply Null Sphere or Rift Walk again and auto attack until they are finished. The combo is short and sweet, and deadly.

9/0/21, that Spell Pen. works wonders and the Utility tree loves Kassadin.

Magic Pen. Marks
Mana Regen. Seals
Cooldown/level Glyphs
Magic Pen. Quints

Cleanse
Hard CC hurts Kassadin A LOT, but with Cleanse all of your problems just disappear.

Flash
Sometimes one Rift Walk isn't enough, having Flash as a back-up has saved me SO many times.

Other good options for Kassadin:

Teleport
Teleport is an excellent skill for Kassadin, and I use it sometimes, but it depends on my team.

Ignite
Ignite is good for a particular mastery tree, just not the one I prefer to use.

The hardest part of playing Kassadin is in the earliest stages of the game. He starts off rather squishy, and unlike most mages he is a melee mage, so farming can be difficult. Zoning is important; keep the minion tide close to your tower, that way you're only steps away from safety.

Kassadin makes his bankroll off of kills, he doesnt need to last hit every minion ever. Do it when you can, but dont take risks in the lane; at least until you get Rift Walk.

Once you have Rift Walk the game becomes MUCH easier and more fun because now Kassadin has escapability and initiation capability.

The difference in my build is a small one, but a huge one at the same time. But I'll start from the beginning.

I start with a Mana Crystal and 2 Health Potions



I don't start with Doran's Ring like many others would, especially after they raised it's price. This way I get plenty of mana and 2 health potions, this way I can stay in the lane much longer and get Rift Walk ASAP.

Naturally the next buy is Catalyst and Boots.



I don't get Mejai's right away, I secure my sustainability and escapability first; but fret not, Mejai's is next.



Now we're ready to kill. Remember the combo and come back after you've gotten 3 or 4 kills/assists.



How easy was that? Super easy yes? Of course it was! Now pick up your Rod of Ages and you Banshee's veil if you can afford it. The Rod of Ages is the difference maker in this build, that boost in HP means so much. Kassadin is able to be durable, why not let him? Now go back out there and be Kassadin again.



The finishing touches to an unstoppable force. The Archangel Rabadon combo is too good to cut; a staple in AP Kassadin. Veigar, eat your heart out.