Kassadin is a mage assassin, and he's very good at his job. He has a silence, a slow, and a flash in his arsenal, and each of those hit very hard as the game goes on. He gets in, gets out, and if you play smart you'll never die.

Void Stone reduces all magic damage by 10%, which is why he lanes so well against other mages.

Null Sphere is your primary tool for harassing in the lane. It hits hard and silences so the enemy laner can't respond with abilities.

Nether Blade is the last skill you put any points into, although I've heard there's a decent DPS build for him out there. But this is not such a build.

Force Pulse deals a large amount of damage in a cone and slows all targets hit, this makes it very hard for your prey to escape. Also useful for farming in lane.

rift walk Rift Walk is what makes Kassadin special, what makes him such an amazing assassin. The cooldown is very low, starting at 6 seconds without CDR. It also deals damage on arrival, which is important because that damage is what will secure most kills.

Teleport is a great spell on Kassadin. Your first time going B is basically free as it relates to in-lane time, and later in the game it can help you get into position quite easily using your wards.

Flash is just a really good spell. Yes Kassadin has one built in but he doesn't get it until level 6 and it's not very spammable until mid-late game. Great way to counter an early gank, and great as a backup if your Rift Walk is on CD or you're OoM.

Start with Doran's Ring. (475 G)

Doran's Ring is very efficient, it grants 15 AP (which is huge early) 100 HP and 5 per/ 5 second mana regen. This means that Q spamming can be done more regularly without going OoM, said Q spam will hit harder than normal, and of course more HP is less death. This should increase your initial lane time by approx. 1 minute before pressing B.

sight ward. (860)

Return faster, stronger, and better after buying the above items. You should have 50 AP, extra mobility, and a ward against the gank.

and boots of choice (1850-2050)

Perfect time to start Mejai's because this is about the time at which you'll start getting your kills; the sapphire crystal will provide much needed mana while building towards either catalyst or tear. Also a great time to upgrade your boots.

catalyst the protector (1785)
or
(1450)

Time to set up either your Archangel's Staff or Rod of Ages which will each carry you through the mid-game while building Rabadon's Deathcap. Which one you take first depends entirely on your play style and the game state.



Finish your big three in whatever order you deem necessary.

SITUATIONAL-

The laning phase is the most important part of the game when using Kassadin, and it is when most new Kassadin players fail because being a melee mage can make farming difficult depending on your match-up. Against other AP heroes Kassadin thrives, because his passive reduces how effectively his opponent can harass, and with Null Sphere he can shut down harassment completely whilst harassing heavily himself. Against other types of heroes the process becomes far more difficult.

Against Ranged Carries (Ashe, Tristana, Miss Fortune, etc.) one must be cautious. Kassadin does not take kindly to physical damage and should avoid it as much as possible. Hit them with Null Sphere whenever you can, but don't go out of your way to do so, it just might cost you. Hug that turret and do whatever is necessary to last hit those creeps.

Against a Tanky DPS (Garen, Jarvan IV, Xin Zhao, etc.) it's easier to avoid trouble, but they shrug off damage so naturally that out-harassing one (assuming competency) is not a likely scenario.

I'll provide an example of each type that you might see when you SoloMid.

Facing Anivia (AP Carry)

Anivia is a great match up for Kassadin, but that doesn't mean you shouldn't be careful. Anivia can flash freeze frost bite harass quite effectively and having a ranged auto-attack also makes it difficult for you to compete. The reason Kassadin is a counter in spite of the disadvantages is because his passive makes up for the difference in damage output and because Null Sphere silences Anivia, forcing her to back up and pop a potion. The AP match up is the easiest, but don't get careless because you'll be in for a nasty surprise.

Facing Ashe (Ranged Carry)

Ashe is a very common (and highly effective) SoloMid and you must be careful against her. Her Volley and her passive can/will deplete your health substantially if you let them. Harassment of Ashe (or any ranged carry) will be done more defensively, hitting her with Null Sphere only when she moves in to harass. Ranged carries tend to be squishy (especially early game) so a couple well placed Null Sphere's will keep her from going super aggro on you.

Facing Pantheon (Tanky DPS)

While the SoloMid melee DPS champs are not common, they are certainly not ineffective. Most casters have a relatively easy time playing around the tanky melee DPS champs like Pantheon that SoloMid every once in a while, but Kassadin has a much harder time. In this example matchup (Pantheon) you have quite a large disadvantage: Pantheon's Spear Shot is a ranged ability that has ridiculous scaling (which increases per level) on AD. This hits you much harder than your Null Sphere will hit him, so early aggressive play is out of the question. His Aegis of Zeonia is a ranged stun that closes the distance between you and him, so this punishes any over-extension on your part and sets up Heart-seeker strike. Your best way to play around this is to last hit with Q when you're receiving pressure and steal last hits with your auto-attack whenever you safely can.
Just play D until you get Riftwalk, then call in your Roamer or Jungler to roll through for a gank.

A good Kassadin can lane against any champion in the game, but one can only be a good Kassadin if he/she knows how to play against the variety of champions that can/will lane against you.