The main thing you want on Kennen, is a lot of damage, but a maximum of durability as well.
Why these items: You can stay into your lane for a huge amount of time so you are free to farm and also, boots so if there's any ganks incoming OR if you want to do some kind of early ganks. Some people start with Doran's Shield but I don't find it really good against another AP Carry mid lane.
Other Starting Items:
Why Doran's Blade? Isn't the champion AP?: Doran's Blade is extremely useful in laning phase, since Kennen doesn't have a good sustain from his stats nor his spells, Doran's Blade will helps a lot since you need to auto-attack a lot of times with Kennen, also it adds more damage to your Electrical Surge's passive, which is nice to harass your opponent. Those are not the only reasons tho, it will helps you last hit by a lot since Kennen doesn't have a high base Attack Damage.
Mid Game Items:
Why these items: You will choose Mercurys Treads if they have LOADS of Crowd Control, otherwise, get Sorceress Shoes so you hurt a lil bit more. Will of the Ancients will provide Spell Vamp, which means you can stay at your lane forever and can win several fights because you will always have some heal remaining at each fights.
Late Game Items:
Why these items: If you're not buying a Zhonya's Hourglass as your 3rd item, then you better turn to AD. No but seriously, Zhonya's is the best item to initiate fights if you have no one to initiate, Flash + Ultimate + Hot Keys to activate Zhonya and there you go. Rabadon Deathcaps for a lot more of Damage obviously, Void Staff to pierce enemies' magic resist and Crystal Scepter to have more durability and more damage as well and really good to chase someone like a poor and lonely Soraka..
: Useful against a long duration hard CC like suppression (Malzahar, Skarner and Warwick). Can upgrade it into
: Can be a good defensive and offensive item at the same time. the range of its aura fits with the ultimate AoE range so it reduce the Magic Resist of everyone in your ultimate. Helps a lot when you have another Magic Damage dealer in your team.
: Not always a good idea, always opt for more damage and defensive items at the same time like Zhonya's Hourglass, Abyssal Scepeter or Rylai's Crystal Scepter. But might be very useful if you're the one carrying everything. However, if your ultimate is down, when you will revive, you will be completely useless and easy to kill.
: Very good against a CC team, especially if their team is more based on knock ups or things that cannot be negated by a
: Nice damage item, the passive doesn't scale too well with Kennen's kit but the stats are pretty good.
: The best enchantment with Kennen since you will have a lot of mobility.
: Really useful if you are the losing team of if you have teleport to get back into the fight!
Note: Remember to buy
at some moment in the game (i.e: some teamfights are preparing mid; baron is up) or, when you have all your items, get them as soon as you can, by starting with