The following guide might be lengthy, so bare with me and please read it. If you don't want to, then please don't bother voting or commenting, I prefer to get real feedback. Just move on to something else.

I like to play Teemo as a hybrid for a couple of simple reasons.
1: It forces enemy champions to build armor AND magic resistance, because more health won't change much, and I deal lots of each. If they don't know how to adjust builds to deal with incoming damage, then you really need to finish your ELO climb.
2: It allows you to get solid amounts of damage from your mushrooms without making your basic attacks useless except for the damage of time effect.
Now that you know why I play him as a hybrid, let's get down to business!

Surge
Surge is a buffing spell that gives bot ability power and attack speed. At the end of the game, Teemo's attack speed should be close to maximum, and at maximum with this spell. The ability power also gives more damage per hit from toxic shot.

Flash
Flash is a way to escape from tough situations when move quick won't help you. It also allows you to pop into a team fight unexpected when you aren't close enough to help immediately when it starts.

Marks
Why do I build Magic Penetration? The point of the marks is to ensure that I can still handle champions who are building magic resist early because they notice I have a decent amount of Ability Power. It also helps deal with the people who keep the magic resist late game, especially tanks.
Seals
Why do I build Health? The point of these seals are to provide Teemo with the health he misses out on in his first session in lane when he has nothing but a vampiric scepter. While the scepter provides great sustain, it does nothing for durability.
Glyphs
Why do I build Ability Power per Level? Teemo's scaling is decent but adding the ability power/level pretty much turns him into a carry enough that he would actually be labeled that way in the pvp.net, though if you really think about it he already is a carry in a way. At any rate, these glyphs proved larger amounts of damage from his toxic shot and noxious traps before he builds his deathcap.
Quintessences
"...Is this guy insane? 2% spell vamp, 2% lifesteal, and 1.5% movement?"
No, I'm not crazy. The lifesteal and spell vamp are both important, and neither should take higher priority, because this build is supposed to hit through the weak defenses. Boots are built slightly later than normal, so I compensate with the single remaining Quintessence slot, my masteries, and skill order.

Mental Force- This provides some additional damage from my blinding dart and toxic shot in early laning.
Sorcery- 4% CDR gives me more flexibility with initiation and escaping with Move Quick, plus it allows me to harass with Teemo's Blinding Dart more often.
Arcane Knowledge- This will give me more damage through the magic resist of tanks and carries who build things like the Maw of Malmortius.
Butcher and Demolitionist- After Arcane knowledge, the next option for many people is havoc. HAVOC IS ****. 0.3% extra damage is pretty much nothing. Instead I take Butcher and Demolitionist to give better farming and pushing power.
Vampirism- Great sustain and survivablity bonus in a team fight. Between my masteries and runes alone, I walk in with 5% lifesteal.
Blast- Blast provides better damage scaling for Teemo's Toxic shot, Noxious Trap, and Blinding Dart.
Expanded Mind- To put it quite frank, Good hands is pointless to a champion who uses mana when expanded mind is a better option.
Summoner's Insight- Allows me to use Flash more often, which is much more significant than a 1 second cast time reduction to recall.
Swiftness- Between Swiftness, my movement quintessence, and my Move Quick Passive/Active, I have much more flexibility as to when I get my boots.
Transmutation- Great sustain and survivablity bonus in a team fight. Between my masteries and runes alone, I walk in with 5% spell vamp.

Blinding dart will prevent retaliation with basic attacks when harassing in lane. For this reason, I take a point at first level.
Your most important skill is Toxic shot, so take it next. Then move quick.
Even though you take a point at level one, You SHOULDN'T necisarily max blinding dart first. Its priority should be based on how many enemy champions are building for attack damage.

This teemo build is meant to play as a carry, so middle lane or soloing the top lane is best. In the case of being in the bottom lane, the best partner is Soraka. She passively gives Teemo magic resist (his defensive weak point) in addition to providing additional sustain.

First buy: Vampiric Scepter
The vampiric scepter provides great sustain as long as you continue to harass your opposing laner(s). If they out harass you, you may lose its benefit, as you won't be able to farm enough.
Second buy: Madred's Razors
Madred's Razors provide you with great farming up until you build them into your final item at the end of the game. They also promote higher physical damage output, which isn't usually the strong point of this build end game.
Third buy: Mercury Treads
Mercury Treads provide some magic resist, which is the one big weakness of this build. The tenacity also gives him a better chance of escape when flashing out of a rough spot.
Fourth buy: Hextech Gunblade
The gunblade gives both lifesteal AND spell vamp, which are both very important in this build. It also gives ability power and attack damage, which increase your damage output tremendously. To top all that off, It has an active that gives you more damage in a team fight. Possibly one of the most important items in this build.
Fourth buy: Phage
By the time you finish the gunblade, the effectiveness of your health seals has been tiny for a while. Phage gives you part of the frozen mallet, which you buy later, as well as supplying a solid amount of health.
Fifth buy: Malady
Malady is a great item for every Teemo. It's effectively providing 42 Attack damage, even if it happens to be magical instead of physical. It give Ability power, for more potent toxic shots, and more attack speed, so you can apply it more often. Top it all off, it gives a magic resistance reduction on hit, which makes Toxic shot hit much harder.
Sixth buy: Frozen Mallet
At this point you should be starting to feel a bit squishy again. Frozen mallet will provide the health you need at the end of the game, as well as making the slow on hit effect an automatic instead of a 35% chance.
Seventh buy: Rabadon's Deathcap
This item, as we all know, is the staple of a mage. It gives 140 Ability power, and to top that off it multiplies your ability power total by 1.3, which dramatically increases Teemo's damage output.
Final buy: Madred's Bloodrazor
Between the bloodrazor and malady, you should be able to tear through any tank you come across. You also get higher attack speed and attack damage which make you flat damage to squishy targets more useful.

Pros:

Deals good damage to all champions, no matter how they build their defensive stats
Still gets kills and asists while dead
Can go through a team fight and come out fine if played carefully
Has great scouting ability from "shroom warding"
Has above average speed
High attack speed

Cons:

Squishier than most champion builds
Has to play defensively at the beginning of the game in order to carry properly
Has to enter team fights cautiously to avoid being a target
Shrooms cost a decent amount of mana
Only 4% cooldown Reduction

Summary: This build deals large amounts of damage and is slightly squishy. It annihilates champions despite their tankiness and specialization. It has enough health return from lifesteal and spell vamp that it sustains well and comes out victorious in 1v1 combats.

Problem: This build is basically only going to work its best if Toxic Shot procs spell vamp, which it doesn't. So basically, if that changes, this guide will become more applicable. Also, why Riot? WWWWHHHHYYYY?!?!