Hello all! My name is Omicrex and this is my first build/guide ever! I'll be explaining my method for playing Teemo. Any comments and tips or welcome below!

Alright so this build, as you may have seen, is about dealing as much damage as fast as possible with your Toxic Shot. You'll be building a hybrid Teemo so AP and AD are the priority here.

Greater Seal of Alacrity - With +0.76% Attack Speed you'll attack 6.84% faster right out of the gate (which is good for early kills for this expensive build)
Greater Mark of Might - As you get into the end game you want your darts to have a little extra oomph so 21.06 extra damage is the best you can do with these runes
Greater Glyph of Strength - +0.28 Flat Attack damage gives you some extra early damage, of 2.52, to boost the chances of an early kill/assist
Greater Quintessence of Destruction - +2 Armor Penetration, and +1.3 Magic Penetration each helps your damage get through when they first start building some defense

Okay, so I'm gonna suggest you go with a 20:1:9 for your mastery trees
Summoner's Resolve
and

Summoner's Insight just b/c you're using Heal and Teleport and want to use them as much as possible

Offense:

Mental Force - Some extra AP at the beginning (and all of your abilities use AP)

Sorcery - +4% Cooldown Reduction for more Blinding Dart and Noxious Trap

Alacrity - +4% Attack Speed for..... well.... faster attacks silly :P

Arcane Knowledge rather then Weapon Expertise - Because you want to damage from your Blinding Dart, Toxic Shot, and Noxious Trap to go through more than your basic attack

Blast - For 1 extra point of AP each level up!

Archmage - Increase your AP by 3.75%!!!

Utility:

Expanded Mind - 12 extra total Mana for each level up

Meditation - for +2 mana Regeneration since you'll be using a lot for team fight and placing mushrooms throughout the map

Swiftness - Since Teemo is going to be squishy, you'll want the bonus 1% Movement Speed to get away from those fatal encounters

Runic Affinity - THE BLUE BUFF IS YOUR FRIEND!!! Use the extra time wisely! Even pick up the occasional Red Buff for even more DOT in addition to your poison

Early Game:
Doran's Blade - Sure it takes all your starting gold, but you get +10 damage, 80 health, and 3% Life Steal!

Boots - movement is key for getting out of those fatal encounters mentioned above

Dagger - Attack faster!!!!

Fiendish Codex - +30 AP, 7 Mana Regeneration, and some early Cooldown Reduction! (10%)

Berserker's Greaves - Time to move and attack even faster!

Stinger - Even faster attacks and Cooldown Reduction!

Nashor's Tooth - An overall increase in your Attack Speed, Cooldown Reduction, AP, and Mana Regeneration!

Guinsoo's Rageblade - Alright, so here is where you as the player who is following this build/guide get a choice. Do you want more AD right now? If yes Pickaxe is for you. Do you want more AP? If so Blasting Wand is yours to buy. Once you purchase one of the two items, go out, get some more gold, and buy the rest of this item recipe.

Vampiric Scepter - At this point you're doing some damage, kicking butt, and taking names right? If so this addition Life Steal will let you continue your streak. If not, may it will help you out! :)

B. F. Sword - Time for a damage boost! Grab a B. F. Sword and show'em what you've got!

Phage - Are your enemies hitting too hard, or is your health too low? (Probably the latter of the two) Grab a Phage from the shop to get a slight AD boost and some extra health.

Giant's Belt - Is their damage still too much for you to take? Grab a Giant's Belt to withstand just a little while longer! >.<

Frozen Mallet - Okay, with this mighty Frozen Mallet your little Yordle body will get more resilient with 700 extra health. Also! The passive effect of slowing the enemies when used with your Move Quick Skill will allow you to keep up with them while dealing as much damage as possible. OR!!!!! Let you run away if you are going to die!

Well, here we are. The final item(s):

Nashor's Tooth - Now, for the point of this build a 2nd Nashor's Tooth is a must for DOT, Mana Regeneration, and Attack Speed.

But as the game progresses you might want to consider some alternate options:

Thornmail - The highest level of AD Armor in the game as of yet. 100 Armor, and as a Unique Passive: Return 30% of damage dealt from enemy standard attacks as magic damage back to them! This is helpful when the enemy team has a high concentration of AD champions.

Alternately:

Force of Nature - 40 Health Regeneration, 76 Magic Resist, and some more Movement Speed! It also has a Unique Passive: Restores 1.75% of your champion's maximum health per 5 seconds. This here is a great item to buy for when the enemy has some powerful AP Champs, and you can run away from your own demise, heal up quicker, and return to the fray to gain some kills for your team.

And finally, for those long games or just for shiggles:

Elixir of Agility - Extra Attack Speed and Crit. Chance for 4 minutes

Elixir of Brilliance - Extra AP and 10% Cooldown Reduction for 4 minutes

Elixir of Fortitude - Extra Health and AD for 4 minutes

Oracle's Elixir - 750 Radius of Stealth Detection. (useful when fight other stealth Champions:
i.e. Teemo, Twitch, or Evelynn

Alright, so at the very beginning of the game you CAN choose between the Toxic Shot and the Blinding Dart. If you decide to go with Toxic Shot first use this for practically the same game:
Anyways, I like to use the first Skill Sequence b/c it allows me to play more of a support role very early game by saving my allies from a few standard attacks from the enemy champs with Blinding Dart.

Afterwards, I start down the path of no return..... I go with Toxic Shot twice to rack up the DOT. Then I go with the movement speed.

Now, this is just my favorite way to choose my skills, and you don't have to follow me word for word, but I just feel that it goes pretty much hand-in-hand with the item build for maximum potency.

Once you reach Level 6 start putting up your Noxious Traps for ganks, escape routes, or to help with neutral minions. I like to use my extra mushrooms to help get Blue Buff or the occasional Red Buff.

Time for the spells!
Heal


Pros:

Restores Health to you
allied champions
allied minions
you can stay in lane longer as health reaches critical

Cons:

lowers the effectiveness of other healing abilities until side effect wears off

Teleport


Pros:

you can teleport to any allied minion, tower, or champion minion
i.e.

Heimerdinger's H-28g Evolution Turrets, or Teemo's Noxious Traps
can be cancelled mid spell by pressing "F" a 2nd time


Cons:

if the telportation is cancelled mid casting, the cooldown is still activated

Other Options:
Ignite - For Extra DOT and those of you who are more blood thirsty than I am! XD
or

Flash - Feel you aren't moving fast enough to run from or to the enemy? Why not Flash!?

Just a side note;
If you do decide to follow this guide people might want you dead....

1) Most people don't like facing Teemo
2) Your DOT with Ignite and Red Buff in addition to your Blinding Dart makes you a deadly support that they will focus