Kassadin is a high burst Ap assassin that excels at killing squishes/mages. Kassadin is also one of the harder champions to catch and kill thanks to his Riftwalk. Unlike some champions, Kassadin relies on kills and assists to get the gold for his items, Im not saying you should not farm creeps or last hit, Im saying you should be more focused on killing your enemy(if possible) and that is why I ALWAYS take mid lane with Kass.

Pros:
Very Hard to gank/kill
High burst damage
Strong ganker
Relatively Simple to play

Cons:
Squishy (with my build)
Low mana pool
Weak before he hits lvl 6

The runes can be of your choosing, but typically I run
9x Greater Marks of Insight
9x Greater Seals of Clarity
9x Greater Glyphs of Warding
3x Greater Quints of Insight
I use these runes for a rounded Kassadin.

Marks of Insight for the extra damage, can be switch for flat/per lvl Ap (Your choice)

Seals of Clarity for some nice mana regain. (since Kass has a low mana pool)

Glyphs of Warding for extra protection against AP caster (These runes work well with Kassadin passive)
Quints of Insight for some moar magic pen
AGAIN this is just my rune preference feel free to use whatever works for you.

For Masteries I take the normal 9/0/21 for most AP casters. Nothing new or amazing...

I usually get a Dorans Ring for my first item but if you feel confident, go ahead and grab a Sapphire Crystal and 2 hp pots. On your first back grab yourself Tear of the Goddess (TotG), if you are short on money just buy your boots. Once you have Sorcerers Boots and Mejai's you can gank effectively. I start on my Rabadons Deathcap as soon as I have my Core (Sorcerers,Mejai's,TotG) and finish up my build with Void staff and a defensive item (Banshee,Frozen Heart depending on the situation)

Null Sphere - Kassadin fires an ethereal bolt of void energy, dealing damage and silencing the target for a duration.
Kassadin fires a void bolt, dealing 80/130/180/230/280 (+0.7) magic damage and silencing the target for 1/1.4/1.8/2.2/2.6 second(s).
Cost
70/80/90/100/110 Mana
Range
700
This is your main Harassment tool. Use this whenever your enemy comes within range and to stop channeling enemy ulti.

Nether Blade - Passive: Kassadin's basic attacks restore Mana. The Mana restored is tripled against enemy champions. Active: Kassadin's basic attacks deal additional magic damage.
Passive: Kassadin's basic attacks draw energy from the void, restoring 8/11/14/17/20 Mana each hit. If the target is an enemy champion, the Mana restored is tripled to 24/33/42/51/60.

Active: Kassadin charges his nether blade for 5/5/5/5/5 seconds causing his basic attacks to deal an additional 20/30/40/50/60 (+0.15) magic damage.
Cost
25/25/25/25/25 Mana
One point in this early is all you need, helps with keeping your mana up.

Force Pulse - Kassadin draws energy from spells cast in his vicinity. Upon charging up, Kassadin can use Force Pulse to damage and slow enemies in a cone in front of him.
Kassadin draws energy from spells cast in his vicinity, gaining a charge whenever a spell is cast near him. Upon reaching 6 charges, Kassadin can use Force Pulse to deal 60/115/170/225/280 (+0.7) magic damage and slow enemies by 30/35/40/45/50% for 3 seconds in a cone in front of him.
Cost
80/80/80/80/80 Mana
Range
400
AoE slow with high nuke capabilities YES PLZ!

Riftwalk - Kassadin teleports to a nearby location dealing damage to nearby enemy units. Additionally, multiple Riftwalks in a short period of time cause them to cost additional Mana and deal additional damage.
Kassadin teleports to a nearby location dealing 60/90/120 (+0.5) magic damage to surrounding enemy units. Additionally, each subsequent Riftwalk within the next 7 seconds costs 100/100/100 additional Mana and deals 60/90/120 additional damage.
Cost
100/100/100 Mana
Range
10000
One if not the best get-away skills in the entire game, use this to close the gap between you and a kill or widen the gap between you and an attacking enemy

So kassadin basic combo is: R->Q->E->ignite and if that doesn't kill them then follow up with another riftwalk. (When in doubt RIFTWALK!)

Overall Kassadin is an immensely fun AP Assassin who excels at melting anyone dumb enough to cross you. And ending with that happy note, just a few parting tips.

Play relatively conservative pre level 6 unless you feel like you can get an early kill.

Don't be afraid to bully the enemy, don't let him in on the creeps, if he does get close blast him with your Null Sphere.

Again, always try and go mid. If not, go with someone who has a stun/root.

Thats all I can think of right now. Thanks for reading and happy LOLing.