Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
This build gives you enough AP to do some heavy hitting with your abilities, while giving you respectable AS and AD so that you can stand alone as a 1v1 champ, and not have to run away when all your abilities are on CD after a team battle. It makes Kennen into an even better chaser than he already is, and makes him like a three way love child of Ryze, Nunu, and TF: Ryze in the sense that he has a lot of nukability and this time it's always going to be aoe, Nunu in that he needs to be in the middle of the opponent to do maximum damage (although Kennen can be pretty squishy, so you need to pick your time to dive in wisely) and TF in the sense that he can use his mobility, AS, and AD to hunt down isolated champions (of course in combination with his sizable AP).
As to details on the build order, I would suggest an amplifying tome to start, along with a health pot, but a dagger and a health pot would work as well. I'd also recommend buying you level 1 boots, then Stinger, and then finishing your boots (usually swiftness, but I'm a huge fan of Mercury's Treads against caster heavy teams) before you buy your Nashor's Tooth. I tend to wait until I can buy my Fiendish Codex outright for the CD reduction and AP, as the mana regen from Mekki Pendant does nothing for Kennen. Then it's your choice, I usually get Rylai's asap at this point for the survivability as much as the AP. Mejai's is also useful at that point, so you can start building stacks. I tend to leave the slot I have reserved for Mejai's open for situation items.
A possible variation is to sub out the Mejais and put in a Phantom Dancer, or even Banshee's Veil if the other team is caster heavy There's nothing like Ezreal's ult doing absolutely nothing to you, and you laughing in his face while you and your team chop him up into little tiny pieces. That Mejai's slot usually functions as what i refer to as my flex spot, which I can use to adapt to what the other team throws at me.
Feel free to leave feedback, every build can always be better based on different situations and I'm new at writing up builds. Good luck, have fun, and DON'T BE A FEEDER! =)