First of this build is not for a brand new player: If you do not know what CC, farm, or other buffs or other useful terminology means, please consult another guide, I will not walk you through LOL 101.
Now that that's aside, the purpose of this guide is to teach you how to play Kennen. Kennen has an interesting role in the pantheon of champions available in LOL. His combo can stun whole teams, as well as dealing huge amounts of damage, but is highly dependent on the whole combo actually landing. This guide will help you in teamfights, laning and mid game.

Each of the builds above demonstrates a combination of runes that could be played with Kennen. The first is a balanced early game domination set, with flat hp Greater Seal of Health for a little bulk, ap for increased damage Greater Quintessence of Ability Power and mp for a little late game damage boost Greater Mark of Magic Penetration. The second set is for early game survivability. I KNOW most people will not like full hp runes for anyone, let alone kennen, but this is if you are truly struggling during your early game. The extra hp will keep you alive and save you from burst damage. If you are have these, you may also consider alternate starting items (see items section). The last set of runes is for when you are totally confident you can faceroll your opponents. Greater Mark of Magic Penetration makes a reappearance along with Greater Quintessence of Magic Penetration for lots of magic pen. Glyphs and Seals provide early game ap to start you off.

Ahhh, items, the core of all builds. Each build here has a different path for a different situation. For almost all situations, I would recommend starting with Boots and 3 Health Potion. This gives you speed to avoid nasty pokes, as well as auto-attacks, which can be surprisingly deadly early game. The only time I would recommend a different start would be if you know you are facing an extremely strong early lane of Garen and Singed proportions. A good example of a mid like this would be someone like Annie, who can hurt you bad early on. In this case, I would suggest going with Doran's Shield: This gives more health to survive bursts.
After your first B, this gives you some options: Ask yourself these questions:
1. Did I die to get here? 2. Do I have a kill/first blood?
If you answered yes to 1, then you should probably follow the second build. It is for situations where you are outmatched, and must build bulk to be effective. Buy a Giant's Belt if you can, or buy some more pots and head back to lane. In this situation, I would suggest Mercury's Treads, especially if facing said Annie or other champs heavy on cc. The rest of the items provide defenses against various threats.
Order here is by no means set, and items can be substituted in. Facing Ashe, Master Yi and Caitlyn? Throw in a Thornmail before any other items. Laning against Malzahar? Throw in a Quicksilver Sash. (Seriously that item is awesome. Free Cleanse AND MR? I'll take it). Flexibility is the key, and those items are just a few of the ones that can help you're situation.
If you answered yes to 2, you might consider going down the facerolling path. This build focuses more on damage output than survival. A Rabadon's Deathcap early delivers huge damage potential, and Lich Bane just gives you even more.
***NOTE***
I know Lich Bane gives wasted stats on Kennen i.e. Mana. The stats still make it worthwhile however. 80ap is nothing to sniff at, and mr gives good bulk against other casters. The passive is what this item is really about though. It allows you to deal out some real hurt, as well as sustained dps due to your spammable abilities. Furthermore, Lich Bane diversifies your damage, dealing physical damage. On a team full of ap, this can devastate a team that builds only mr to counter your damage, and it keeps the enemy honest.
***END NOTE***
Anyways, That combo gives huge dps. At this point, you'll find the other team starts to focus you really hard, due to your ninja prowess. This is why the build adds other bulk, which will keep you alive during teamfights. Feel free to put Zhonya's Hourglass in earlier if cc has you down. The active synergizes well with your ultimate, and it provides a large ap boost.
If you answered yes to both or no to both, You should use the first build. It is a versatile build, with sufficient bulk to survive teamfights, and enough damage to stay relevant late in the game.
Note that Rylai's Crystal Scepter is built before Hextech Revolver. This is because Rylai's will allow you to survive the bursts so typical of mid lane champions. While Hextech Revolver may give better sustain over long periods of time, These must be extremely long periods of time, for at low levels, the sustain at lower levels is negligible. A full combo at level 5 offers just 35 health back, even before damage reduction due to MR. Spell Vamp is excellent, don't get me wrong, but it is not useful early in the game. I almost never play a game without Rylai's it's just an awesome item. Good health bonus, good ap bonus and a nice cc (if soft) for a passive is just to good to pass up. It fits Kennen's playstyle perfectly, and I would recommend obtaining it 99999/100000 times you play him.
Remember: All these choices are not set in stone. If another item is more beneficial in the situation, don't hesitate to mix and match from different builds. There are some items however, I would caution against having. These include:
Abyssal Mask: While a good item for a caster tank or offtank to have, leave it to them. You have better items to pursue.
Archangel's Staff: Why? Why would you do this?
Anything with CDR: You don't really need this. Your Thundering Shuriken has a low cooldown, and spamming your W+E will just leave you gasping for energy when you really need it.
Rod of Ages: While an excellent item, the main bonus on this is the mana, which Kennan does not possess. The health and AP are nice, but take to long to come to fruition.

Thundering Shuriken is your bread and butter. Know it. Love it. Practice with it. Make love to it. I don't care what you do as long as you love this skill and use it to it's fullest. Thundering Shuriken is an excellent tool for farming, poking and doing lots of damage to squishy carries. It essentially gives you a bigger last hit for farming, as well as allowing you to last hit multiple enemies that may be low on hp. You can also zone or harass with it during lane phase. It does large amounts of damage for early game, and can keep opponents wary and allow you to farm easier. It is a skillshot so practice with it before using it.
Electrical Surge is another good poke, as well as a critical part of your combo. When you have full stacks of it, you can last hit or poke an enemy champ for some bonus damage, then activate the ability to deal some extra damage. It can also work in conjunction with Lightning Rush to clear entire minion waves in one go.
Lightning Rush is an excellent escape tool, as well as another part of your gamebreaking combo. It ups your speed quite a bit, and allows you to move through minions and champions. Unless starting a fight, avoid using this for damage, as it has a longish cd and uses quite a bit of energy.
Slicing Maelstrom is the final part of your game breaking combo. It hits hard, and stuns large groups of enemies. It also works well with Zhonya's Hourglass, where you can pop your ult, stun everyone, then protect yourself from retaliation with the active.

Laning:
Kennen is a wonderful laner due to his ability to farm effectively and stay out of trouble. His Thundering Shuriken is an excellent farming tool, ensuring he will rarely be starved. He can also stun, even early on, allowing for good support from Junglers. He can also spar with most champions excepting annoying, high powered early laners such as Caitlyn. He also does fairly well against lane bullies such as Garen or Talon because he can farm effectively from a distance, and his threat of a stun. I would suggest laning Kennen mid, but if some pub decides you don't deserve it and won't budge, Kennen can be effective in any lane.
***NOTE ON JUNGLING***
Don't do it. Please. You are wasting the laning ability of an excellent champion. PLus he kinda sucks at jungling for the most part, so just don't.
Mid Game:
Kennen's mid game is fairly good. He can show up in a lane and completely decimate the opposing team. Until he gets Lich Bane, he isn't much good for hurting towers, but once he does he can really burn them bad if left unchecked. Kennen can also often just lane right through mid game if your team has everything under control.
Late Game:
Kennen's late game is pretty fricken' awesome. He can decimate the enemy team when the team is working well together. The important part of the late game is the teamfighting.
In a teamfight, Kennan should not be the initiator: even with some bulk he is still fairly squishy. Rather, he should go in right after the initiator. This allows for him to fire off his combo early, but without the huge focus that ends up on an initiator. When comboing, pop your Lightning Rush and charge through their squishies. activate your Slicing Maelstrom while in their formation and use Electrical Surge to stun them. Use your Thundering Shuriken on the most heavily damaged squishy and keep spamming it until they die. That's pretty much all there is to it.
Lane Match-ups to avoid:
Caitlyn: It depends on the player, but a good Cait will make farming very hard. She is possibly the only champ that can starve you of farm.
Fiddlesticks: Goes for anyone really. You can stop him, but he'll do a billion damage while you try.
Galio: Just plain annoying. Builds super bulky and has a great harass.
Kassadin: Impossible to catch+silence=:(
Ryze: This *** can stop your combo outright with his snare.
Sion: Very hard to kill and has great burst. Be wary.
Swain: He has incredible sustain, and can therefore harass you hard. He also has quite a bit of CC, making him a tough matchup.
Vayne: Screw Vayne. She does way to much damage and is way to easy to play. She should have nerfs coming her way, but like Tryndamere, Riot's staff is so much in love with her that they 'nerf' her and really just make her better.

Kennen is an exemplary champion. He can deal tons of damage, deliver hard and soft cc and make or break teamfights. Use him well, and you shall rise up the ranks and stand glorious in victory! May you stand victorious over your foes!