Not Updated For Current Season
This is the way I play Teemo, and this Teemo works great with any other assassin or mage, like Lux.
You may find strange that I set armor pen, but it's not, otherwise you deal no damage at all early game, and it can help piercing malphite's shield or things like that.
Once you get the malady, you will deal high damages because of your poison, and even more with the sword of the divine, not to mention the awesome 30 armor pen as its active ability.
Some lifesteal to keep safe in teamfights/ helps farming/ good in any way since you have few life.
Wit's End, magic damages, with the malady it will hurt really bad, even with more attack speed so the sword of the divine passive will trigger even more often.
After that, the Hex Revolver, you can also buy the lantern but I couldn't find it in this website so make a choice, both are great.
Take the blue buff when possible, red is less needed, let it to your carry ( no, teemo is NOT a carry ), also ask your mage if he needs the blue. You might try to steal the ennemy buff, as mushrooms eat your mana. If you're REALLY annoyed by the mana problem buy a Nashor's Tooth instead of a wit's end.
Play safe, always back at any sign of gank, I recommend playing lane, solo teemo can be crushed easily by characters like Morde, Morgana ( anti-poison shield ).
Stay far from the teamfight and finish fleeing ennemies, teemo is an assassin, not a fighter. If you're focused you will last no more than 4 seconds.
Perhaps you want to take exhaust to make sure you won't get raped by a Xin or a Jax ==> DO, NOT.
Ignite is mandatory. Ennemy walks on your shrooms, get out of invi, ignite, and auto-attack. DO NOT use your dart unless you're SURE to kill him. The dart has a better ranger but does far less damages than your auto-attack with this Teemo, so just hit him without a stop and if he becomes aggressive then yes, use your Q and here we go.
Sword of the divine is a great help against Jax also,or any tank that rather stacks armor.
What if the other team gets an oracle? DO NOT PANIC, don't stop planting mushrooms everywhere. They DO NOT give gold anymore so don't hesitate, they will lose time destroying them and you still know where they are, your mushrooms help so much for the ganks, place them at dragon, nashor, golem and lizard, and in EVERY BRUSH, EVERY.
Hope you do well, also don't forget: Never activate your W unless you're in danger, because if you get surprised you'll have to wait 4 seconds for it to be up and these will cost you death.
Edit: Stark's Fervor is also a really good choice. With the malady, wit's end, stark's fervor and sword of the divine, you will have 2.5 attack speed. If like me you got the astronaut teemo, it's going to be " pew pew pew pew " like it never was.