Im BadPiriPiri.
I love teemo, its fun, strong, but i feel the ap build is overused, not viable at all, so i decided to share my build.

Build the core items and build all of the offensives if your team is winning, if you are losing focus on the deffensive items.

Ignite - Use on normal and easy Match-ups
Flash - Required
Teleport - Use against your counters
Exaust - Use when your against a riven or something with early burst

Pros

+ High area damage
+ Tanky
+ Infinite Slow
+ Blind
+ Useful on team fights
+ Great Duelist
+ Pusher
Cons

+ Weak if behind
+ Mushrooms doesn't have damage
+ No Hard CC

Fury VS Sorcery - Fury is better because this is a Basic Attack Based Build
Fresh Blood VS Feast VS Expose Weakness - Fresh Blood.
Vampirism VS Natural Talent - I prefer Natural Talent because it gives me more Damage, but vampirism is better if you are against a Poke champion
Bonty Hunter VS Double Edged Sword VS Battle Trance - Battle Trance!

At lvl 1, try to farm the entire wave, and poke him, then when you got 2 try to fight with him.
Farm as much as you can.
Frozen mallet is your core item as soon as u get it abuse it, you hit him and he can't run away from you, you can kite easily, and you get tanky, then with runaans, you farm like a god, your powerspike is when you buy guinsoo, you will burst everyone, the other items are opcional but wit's end is very good too, you only build ruined if you're winning the game and they have tanks.
When fighting, Kite him, use your Q and use your mushroom on him for dps (damage per second) and the slow.

Stay near to your carriers, and attack anyone who gets close to you.
Or you can try to attack their carriers but that is more risky (Only do this if their carriers are badly positionated).
Use your Mushroom to slow them.

This build is good against teams with a lot of tanks, because of the wit's end and ruined king, and you do a lot of damage on squishis.

Patch 6.23 - Masteries updated
Patch 6.20 - Added