This is my first giude ever so please no flames... Kassadin has been my main for some months now so I believe I believe I know he a bit by now. Kassadin is one of the most powerful AP-nukers in the game in my opinion, something i hope to show you by this guide.
Last... Engish is not my main language so I will problaly mess up some of my grammar and words, but you are here to learn Kassadin. Not to point out my poor english :P

Pros:

Anti-caster
Sick ulti
Epic late-game
Looks like a badass
Has some CC (slow, silence)

Cons:

Meele
Glass canon (without Frozen Heart and ROA/Rylai's Crystall scepter)
People will focus you in teamfight
can't go mid against AD-range (Caitlyn, Graves etc.)

I use Greater Quintessence of Potency for a bost of AP in the start. Becasue of this, I dont need to buy a Doran's ring. I realy dont like Doran-items. mainly cause they cant be built into anything. My glyphs and seals are there for max coldown reduction. These runes + Ionian boots + Frozen heart, should get your ultimate down to under 3 sec coldown. The marks I am using is there since Kassadin can be really mana-hungry, especially if your going to spam your ulti.

My masteries are plain normal Ap-caster masteries. Ap, CD, and mana regeneration. I also take improved flash and ignite. the improved flash also improves Clarity if you take that summoner spell instead of flash or ignite.

I am maxing my Q first, mainly since its not a skil shot, can silence and before Mid/Late-game your E won't be up as often as you want it to be. I am taking my ulti as often as possible... Duh... And taking my first point in W at lvl 7. I am not a big fan of it's active, but I use it to charge up my E and cause of the nice mana regeneration passive. Then I just keep it at lvl 1 untill all my other skills reach lvl 5.

I usually take Flash and Ignite, though i have been using Clarity instead of Ignite my last games. Some people says Kassadin don't need Flash cause of he's ulti. I totally disagree. I love to Riftwalk in - nuke - flash out, and hear the game say: "an enemy has been slayn" :D It is really just nice and safe too have. if your out of mana, have a really fast player after you (Master Yi), or if you want to nuke as I do.

As for ignite: I used to use it, since its good to have if you Riftwalk in - nuke - Flash out and they still have some life left, but I feel like its useless in lategame. Also as you might have seen, I dont use Archangel's staff, this can put me into some really kinky mana situations. Wich is why I have started using Clarity. I really believe summoner spells it's all about what you like and are used too.

I start of whith a mana crystal and 2 health pots. The reason for this is cause if you play smart, the extra mana will let you last hit with your Q and let you stay in lane longer. Also instead of that useless Doran's ring, your mana crystall can be used in your next item. Try to get CAtalyst befoore lvl 8, as its unique active/passive really helps you stay in lane. Recall first time when you have around 950 gold. after the explorer buy shoes. Try to rush to ROA. After you got ROA finish your boots.
My "Core build" as Kassadin is ROA, Ionian boots and Rabandon's Deathcap. After this I try too pick my items after who we are against or what my team needs. I try to get Zhonyas next since it gives armor and a nice amount of AP. Its Active skill is AWESOME on Kassadin. The reason for this is since Kassadins biggest fear is CC, if you get stunned or silent. just activate Zhonyas and wait for the stun/silens to run out.
If the enemy team is to powerfull, buy Frozen Heart before Zonyas. Frozen Heart is a great item on Kassadin cause of its armor, mana and CD.
After this I try to get even more Ap with Void staff and trade in ROA for Rylais.