Begin with Amplifying Tome, compared to Doran's ring you get --v

1. 5 more AP
2. Mana Rain does not matter to Kennen
3. You get double in hp if you take the amplifying tome (Health pot, 150 hp) than Doran's Ring that gives you 80+ max hp.

Then upgrade the Amplifying Tome to Hextexh Revolver and get boots (preferably the same shop) and leave them there until you have acquired a Rylai's Crystal Scepter.

Important to Rylai's is to get Giant's Belt if you are in your lane, Blasting Wand and Amplifying Tome if you win it.
Once you've purchased your Rylai's, you'll buy your Will of the Ancients and Sorcerer's Shoes. Again, if you're underneath, you'll get Mercury Treads if you get bored with Magic damage, and Tabi's if you meet an AD.
Important with the knowledge is to also come from your enemies ALL THE TIME, get Abyssal Scepter if you meet a lot of AP and Zhonya's if you meet AD (You will get Zhonya's late game too).
If you do not already have a Zhonya's from previous issues with AD, you'll get it as the next item after Rabadon (Rabadon, you'll buy if you're leading, ofc.), That will let you get your whole ult. I'll be back with your Late-Game build.

When you are harassing in the middle, you will get Q on lvl 1 and w on lvl two. Do not overdo your Q, and use it when you are in control of the situation. On lvl 2 if you take W, keep track of your auto-attack stacks, which also gives you extra dmg and a passive stack if you attack the enemy with the attack. Then you can have your passive function on lvl 2 and you can get a big take over on lvl 2. Usually it's not good to have E so early, I recommend lvl 4, then it's the greatest chance that you're beaten by a jungle Which has all the attacks.

items:
Boots: Sorcerer's, Mercury Treads, Ninja Tabi's
Will of the Ancients
Rylai's Crystal Scepter
Zhonya's Hourglass
Abyssal Scepter
Rabadon's Deathcap