I'm going to start this off my saying that Kennen is one of the best casters in the game. In early laning phase, he stands out as being easy to farm an harass with. In the late lane phase, he can clear minion waves very easily (and getting all of them as creep score at the same time) using lighting rush and then electrical surge, which makes your enemy laner very reluctant to leave for ganks, while leaving you free to take up a gank without risking losing a tower. After laning phase is over, aoe ults and stuns pave the way to easy teamfight wins

Mark of the Storm (Passive)

Riot actually decided to equip Kennen with a useful passive. While it is much harder to land than other stuns (Brand, Xerath, etc.), you can use it on multiple targets (Even the entire enemy team) simultaneously using your three aoe abilities.

Thundering Shuriken (Q)

This is Kennen's bread and butter in lane phase. It has a short cooldown, a good base, good scaling, and amazing range. If you're somehow being pushed back in lane, it can be used for long range creep farming. While not as useful in team fights as his other abilities, once you get rylais, it becomes a very effective chasing tool if utilized properly.

Electrical Surge (W)

Electrical surge serves a lesser role than Kennen's other abilities while in laning phase, but when the passive is utilized and combined with Thundering Shurikan and the Electrical Surge active, it becomes a very good ganking tool as well as a good harassment tool that keeps your opponent on their toes. In teamfights, it is the third thing you should use, as it only deals damage to enemies who already have a mark of the storm. Activating it with two untouched enemies loses damage potential and your stun.

Lightning Rush (E)

This one is a fun one. Often misused, but very fun. It combines aoe damage with defense boosts and a free ghost. It serves multiple purposes, the foremost important being gap closer, escape mechanism, minion farmer, or (among the best uses) mark of the storm adder. It is extremely essential to farming minions in mid to late game. Combined with electrical surge, it can clear an entire minion wave in seconds.

Slicing Maelstrom (R)

This is what makes Kennen the teamfight winner that he is. Be it the aoe stun or the high damage (210 + 40% ap at level 3), when used properly it can be a huge asset to your team in a teamfight. It is imperative that you already have at least one mark of the storm on all enemies in the radius or you risk them escaping because the storm did not proc the stun quickly enough.

Preferred

Flash

Flash is useful as it serves both a getaway, and a gap closer (the fastest in the game in fact). After the nerf in the fizz patch, wherein the range was reduced, it may not be as viable depending on your play style.

Ghost

I know what you are thinking. "Why bring ghost when lighting rush serves essentially the same purpose." Well, the simple answer is "they stack." When you can burst 600 movement speed, nobody can escape your ult and nobody can catch you. Ghost can both get you a kill or prevent a death.

Alternatives

Ignite

All I can say about this one is that it is a good bit of bonus damage.

Exhaust

Exhaust is useful against ad champions, but does significantly less to magic and on-hit champions. The only reason you should pick this over ghost is if you are sure that you will be mid against a strong ad champion (Caitlyn, Vayne), but this is unlikely.

Surge

While it sounds good, the long cd and minimal AP gain at low levels make it less-than-attractive. If you are the type to 3 man gank mid for first blood, surge can help... but otherwise it's not that useful and late game there are better choices anyways.

Doran's Shield

Doran's Shield is nice because it gives alot of early game sustain and health. Alternatively, you can buy Boots of Speed and three health potions OR an amplifying tome and a single health potion.

Hextech Revolver

Greatly extends your lane sustainability. 40 AP also gives your spells that extra 'oomph' that can determine who wins mid lane.

Sorcerer's Shoes

Movement 2 boots are necessary. Some other builds recommend Boots of Swiftness, but that role of keeping people in your ultimate is better achieved through Ghost. And besides, who can dislike 20 magic pen?

Will of the Ancients

This one (I think) is pretty self explanatory. The group buff 25% spell vamp gives Kennen sustainability rivaled by very few (Maybe vlad, but his late game sucks). There is no alternative, as revovler simple doesn't do enough, and Kennen is not enough of a hybrid for a gunblade.

Rabadon's Deathcap

Rabadon's deathcap is the best item in the game for AP. If you are getting pinned down, opt for a defensive item first (Chain vest, Negatron Cloak). Otherwise, pick it up. The 30% AP boost is absolutely amazing for late game.

Rylai's Crystal Scepter

This item is one not commonly found in Kennen builds. I, however, find it essential to Kennen's durability and chasing power. This item makes your Thundering Shuriken do a 35% slow, which means it is very difficult to escape you.

Zhonya's Hourglass

Between the armor and the 2 seconds of invulnerability, this item provides an amazing amount of durability. Please remember that activating your Zhonya's Hourglass DOES NOT cancel your ultimate, so you can sit there nice and secure while your enemies are struck and stunned by lightning.

Abyssal Scepter

This item is very nice. Not only does it provide you with magic resistance, but it also reduces the magic resistance the nearby enemy champions. Think of it as aoe team magic penetration (Since it reduces the amount of MR when calculating your allies' spell damage).

In summary, this build provides a lot of early-mid game damage as well as late game survivability. Please thoroughly test it before voting.