This is a guide for AP Teemo (with some attack speed).
This build will show you how to get a fairly high amount of AP, so your mushrooms will blow your enemys away! And also give you value in team fights with your attack speed.
In straight AP builds, you have little team fight value, due to lack of attack speed, and in straight on-hits or physical builds, your mushrooms suffer the AP that could make the difference.
This build combines the best of both, and some survivability to keep you alive.

If you choose to downvote, please let me know why!

Runes:
I have to make my skills do as much damage as possible.

I have 3x flat HP quintessence to make me less squishy.

I have Greater Seal Of Replenishment, because without, you might have some mana problems when you tend to spam skills and Noxiou's Trap.

I have flat Ability Power Glyph's for kinda heavy early game damage.

Masteries:
I go 21/1/8 for damage, boosting heal and hp/mana regeneration.

Items:

I start with Amplifying tome and a Health Potion.
I start with Amplifying tome's to boost the damage done by .
When i can afford, i go back and buy boots and a health potion if im losing to much health in my lane.


I then wait untill i can afford Malady before i go back to buy.
I buy Malady to get attack speed and AP. And the passive that deals 20 extra magic damage and reducing the targets magic resistance is a very good combo with my magic penetration runes.


I take Sorcerer's Shoes instead of because i will make magic damage with all of my attacks, And will gain attack speed with other items later in game.


Here i either choose Rylai's Crystal Scepter or Rabadon's Deathcap. Rylai's Crystal Scepter if i take alot of damage in my lane, or Rabadon's Deathcap if i feel like im doing more damage then i am taking. If i take Rylai's Crystal Scepter i follow up with Rabadon's Deathcap, and in reverse, Rabadon's Deathcap -> Rylai's Crystal Scepter.


I either buy Nashor's Tooth or Morello's evil tome. If i feel for attack speed like so many people do with Teemo, I'll take Nashor's Tooth. Else i will go with Morello's evil tome for the ability power. Both of the items gives cooldowns wich i find needed so that i faster can put Noxious Traps all over the map.


In the end i buy Void Staff to get ability power and the magic penetration.


Optional Items if you need defence, is Banshee's Veil, Zhonya's Hourglass or Abyssal Scepter, depending on what situation you are in. If you need defence from Magic attacks, you get Abyssal Scepter. If you need defence from Physical damage, you get Zhonya's Hourglass.
Banshee's Veil is a option if you're facing a heavy CC(Crowd Control) team, like Morgana Amumu Galio.
Zhonya's Hourglass also got a nice ACTIVATE spell. You will turn into gold and will become invulnerable to all damage for 2 seconds.

You can use a hotkey to activate the Zhonya's Hourglass spell fast, instead of finding the item in that small inventory screen and pressing it with your mouse.
The number keys on your keyboard (1,2,3,4,5,6) works as hotkeys to your items, depending on where your item is placed in your inventory.
[1] [2] [3]
[4] [5] [6]
So if you place your Zhonya's Hourglass at the 2nd slot, you can activate it by pressing the number 2 key on your keyboard.

Skills:
I start with Toxic Shot because the basic attacks will deal damage over time with the poison from Toxic Shot. With this i can deal good damage without wasting mana, as i would have done if i started with Blinding Dart.

I boost Toxic Shot and Blinding Dart to deal alot of damage in early game.

I put one point in Noxious Traps every time i can. (lvl 6, 11 and 16)

I put one point in Move Quick at lvl 7 just to gain the extra movement speed if i get ganked, or if my teammates needs help. I then save this skill for last, just becuase it's not needed to be boosted early game, in my opinion.

Summoner spells:
My Choices:
Heal
Flash

I take flash and heal just to be able to go into a fight in early game (i go mid, so 1v1) and do as much damage as possible, just stand there and shoot. When my health got about 2-3 bars left, or as far down as i dare, i use heal. Early game fights often results in that both players do equal damage to eachother, and thats why the extra hp i get will keep me alive, while the other player dies. If the enemy player starts to run after i used heal, i jump after with flash and strikes a final blow with my Blinding Dart or with a basic attack, becuase the poison from my Toxic shot could possibly do the damage required for me to get the kill.
Flash is also good to escape with. Flash over walls, or in direction of a mushroom, where your enemies will be slowed if following you, and this will give you time to get away with .

Optional spells:
Ghost works good with to get away from danger, or to move across the map to gank.

Teleport will allow you to get to your or other's lane quickly since you should be soloing anyways, as well as set up mushrooms in the jungles and along lanes. KEEP IN MIND YOU CAN TELEPORT TO YOUR MUSHROOMS! This is great for ganks.

Ignite is a good spell to make sure you get the first blood. With this makes the enemys hp drain pretty fast. But i find this spell not very needed in late game.

Mushroom placements!

Blue means IMPORTANT! Always have Mushrooms here.
Red is less important, get these locations last.
Green is optional if you play mid. For stoping ganks and doing damage.
There are alot of other places to place mushrooms, but i show the most important and optional places here. A tip it to place shrooms in almost every bush. If a tower is destroyed it's very importand to place in bushes along the lane, to prevent ganks. (I forgot to put that on map, so i write it here instead) Another tip for placing its to take and look where the enemies are moving the most, and place mushrooms there.

5v5

Placing mushrooms a bit infront of your tower will give you time to go and place mushrooms else were on the map, without having to worry to much about the tower. Atleast for a moment.

As i dont play 3v3 with teemo i dont have a placement map for it.
But if you want it, just leave a comment and i will find out and post a picture with placements.