This is a build designed to have you be wherever you need to be to deal the most damage. You will be able to place your traps all overthe map with very little trouble unless the other team has a mobile champion too. You won't be hitting for much but you make up for that with amazing map awareness and some surprizing killing power.

Pros
Extremely funny to see Teemo run around like a crazy person.
You can set your traps pretty much where ever you need to.
Unless you get slowed/stunned/rooted by some means, you can almost always escape danger.

Cons
EXTREMELY squishy.
Tanks, take him down.
Barely useful in early 1v1 and team fights due to lack of attack damage.
Farming dependent.

Passive

Camouflage Camouflage is Teemo's passive and has been severely nerfed. Despite what the icon says, it takes you two seconds without action to become stealthed and you can't do anything. PERIOD. You gain a bonus to attack speed after leaving stealth. People can still run into you so skillshots like Amumu's Bandage Toss can hit you, and then you become un-stealthed and die.
Note: I'm not totally sure if people ca still hit you. I know that minion waves will go around you and sort of expose you but, I don't think that allies will have contact with you as I have had a shaco doing the worm inside me.

His "Q"

A nice long range skill and perfect for last hitting and making sure that physical champions can't do a thing to you. Fun times all around.
Note: It actually has the same range as his attack so, you can use it to determine what you can attack.

His "W"

Move Quickly A nice passive increase to your movespeed and makes you the fastest little Yordle out there. With it you get quite the movespeed buff as long as you stay out of un-contested combat. Or just in the back where you should be.

His "E"

It's an interesting DoT that doesn't do much damage unless you either have a ton of ability power or, you have a ton of magic penetration... Even then it doesn't do much.

His "R"

This is both a really good skill and a really bad skill if used correctly/wrongly. It becomes a small range that lasts for a maximum of ten minutes. At higher levels it can wipe out complete minion waves making him an even better farmer.
Lining bushes makes for a great "safehouse" to recall/initiate from.

Moving on...

Taking [cion=Blinding Dart size=19] Blinding Dart Will help you start off with a good farm game if you know to hit things with it while they are low in HP. It also helps defend against all DPS champions. If you see a ton of fast people, you might want to prioritize Move Quickly Move Quicly over other things. Of course taking your Noxious Trap whenever possible.

Start out with buying Boots of Speed and3 Red potions and haul Teemo's little body to your lane. Stand slightly off to the side and just stealth it out. Head back close to level 6 and buy either a Philosipher's Stone Philosipher's Stone or one of the parts. Continue jumping back and forth helping in lanes as you see fit and buy your Phantom Dancers. By the end of this, you will be just over the Attack speed cap and far over the critical chance cap. Your average speed will be 603 and you can get away from almost anything.

Ghost is used to gain speed. In dicey situations, it can be used as an escape maneuver.
Teleport is just taken to add to the "Where ever you want to be" thing. Drop a mushroom, recall, teleport back!
If you want, you can take these:
Flash is just as good, if not better, than ghost as an escape method. I would recommend taking it in place of teleport.

Once you have your Noxious Traps ready, you want to start laying them at the Wraith camps after you kill them. You will get gold, basically right when they re-spawn and steal gold from any other would-be jungler. You can come back later to pick up the big Wraith. You don't really need any of the runes but if you can get the other team's buffs, all the better for you! Once you start jungling, you want to be running through lanes helping whoever needs it.

I take the move speed Quintessences to, you know, move faster. The other runes are just to help with not being squishy. Dodge chance, armor and a little health regeneration.
That is all... Moving on.

I go with a very odd setup of 15-0-15 so I can have the move speed increase and magic penetration. The slight cool down reduction helps place more mushrooms out faster. The magic penetration and AP per level helps them hurt just a little bit more.

I spent many days on this. Not to refine and tune the guide but, to think of things to make this inside the 5,000 character "minimum". So, since i'm here, I guess there's nothing else to say. But, anyway, this build is slow to get started and fun to play with and you can be almost anywhere you need to be. Lastly, since this is a "for fun" build, I most likely won't be updating this much, if at all. That said... Have fun Teemo-ites!
Note: It took 3 days of adding somewhat unnecessary things and sections to get this to the 5,000 "minimum". Please vote nicely!
EDIT: I have changed it so you have to comment to vote. I really want to know what's to like about this.
EDIT2: Because some people think that this is a useless build, pictures will be up as soon as I possibly can have them. Hopefully in the next few days.

3-6-11: I archived the old build because nobody will see it due to the early downvotes so, this is the completely re-vamped version with many changes to make it better.