First, I will admit now, I am only in Bronze League. But I feel confident enough to display my build that I always use for Teemo.And as such, I ask that if you have an info on certain things told in this guide, please leave a comment or something. If I am wrong, or if an item/ability works differently than what I said, please reply with proof supporting your claim.

Also this guide is going to be pretty...bland. I am not very good at doing this kind of stuff on the computer. I can do the links, but that's about it.

For me, I have had a pretty good success rate at countering just about any champion in any lane with Teemo.
Also I will tell you now, I am very bad at explaining things, but I will do my best. I don't know all the fancy "This champion gains 0.002 Attack Damage per level, and scales at 1.2 Attack Damage per level"...I just know that some champions start off with X amount of Magic Resistance, so I need to get this item, instead of that item if I am going against this champion.

I should also tell you that this build is something that I have been reworking for a couple months now, and it is my final build.
AS well, this will be a very straight-forward guide from here on out.

I do have a request. I can only think of the pro's side of Teemo, so if you can post a comment or something with some cons of playing Teemo, please do so. But try to not be a jerk about it. Remember. The guide is a work in progress. The items are set in stone, but the tips, advice, and other suggestions will be reworked over time until I feel that this guide is as perfect as it can be.

Marks: greater mark of hybrid penetration x9. Great for Teemo since you will be doing both physical damage, and high amounts of magic damage.

Seals: Greater Seal of Scaling Attack Damage x9. Gives you a boost of AD that levels with you. For Teemo these are better than the Greater Seal of Attack Damage because it gives you more AD. Also Teemo is what I call a "Snow-ball Champion". By that I mean Teemo starts off small, but ends up bigger and more dangerous the longer you are in a match, or the faster you get gold.

Glyphs: Greater Glyph of Scaling Ability Power x9. great for the same reasons as the seals.

Quintessence: Greater Quintessence of Scaling Ability Power x3. again, great for Teemo's "snow-ball effect".

With all that you will get
8.1 Armor Penetration at level 18
5.8 Magic Penetration at level 18
20 Attack Damage at level 18
51 Ability Power at level 18.

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Now, because Teemo is a Hybrid Champion, his Mastery Tree should reflect that. I will not go into great detail with the specific talents, but the "Offense" tree has two sides to it. one is for AD champions, and one side is for AP champions. As Teemo it is best to utilize the more functional talents on both sides of the tree. It will give you CD, ASP, bonus damage on hit, some extra critical chance... Everything you need as Teemo.

I find that Teemo is a Champion built to harass his lane early on, and get kills early on.
His Blinding Dart is a great ability. With this build I have it doing nearly 500 damage per shot, plus it Blinds the target for a couple seconds.

But the problem that I have noticed is that between blinding shot and Noxious Trap Teemo CAN run low on mana. Does not usually happen, but it can and it will happen if you are not careful. You never want to be caught without mana.
So to address this problem, I get the Clarity spell.
Also I ALWAYS get Ignite. It has many applications. Burning down champions with too much HP, or making sure that they just don't get away. Ignite combined with the Damage Over Time(DoT) it will help ensure that your opponent doesn't get away with 25 hp.

Other spells can be useful to Teemo. IF you feel confident enough with your mana usage, then get Heal. I don't have to explain how it can be usefull. It heals you. Nuff' said.

Barrier can also be functional, especially if you are fighting against champions with sudden burst attacks. Great to protect you when you are in an emergency. At level 18 it blocks up to 450 damage from any source. Though I am not sure if it blocks damage from Turrets.
Next is Teleport. To me, it is extremely situational. If you like to do surprise ganks, great. or you are falling behind in your lane, and need to get to your turret ASAP... But I dont suggest it for Teemo unless you are Jungling.

Now to save you some time, I only suggest Exhaust and Flash. Both can be used to excape dangerous situations. Exhaust is great. It slows their movement, attack speed, and reduces damage they deal, even from activated items like Deathfire Grasp.
Flash Teleports you a short distance away, in the direction of your cursor. It can ruin a well formed gank, let you escape a dive attack like Aatrox's Dark Flight. It can also be used to catch up to a champion trying to escape.
I dont recommend any other summoner spells.

To me, Teemo is all about Ability Power (AP) Attack Speed (ASP) and On-Hit damage.
The build I give you is focused on that, with some Attack Damage (AD) and Life-Steal (LS).

I do admit that the LS isnt very high, but it is enough to help you survive 1-on-1 encounters, and even sometimes against three or more Champions attacking you. While you want to do your best to avoid facing more than two Champions at once...**** happens.
As a note, with this build, Teemo will have about 170 AD, and about 480 AP, depending on which final item you get.

Now to the items themselves.
As I said all the items focus on AP, ASP, On-Hit damage, with some AD and LS.

Nashor's Tooth. This item is not up for debate. IT gives you AP, AS, and Cool-Down reduction (CD). Also, it does 15 damage, +15% of your AP as bonus damage per hit. Last I checked it makes your basic attacks deal an additional 75 damage.
Couple that with Toxic Shot, which will be doing about 200 bonus damage per basic attack...You create a champion that just has to auto-attack to win.
To build this item, start off with Fiendish Codex then get Stinger and finally Nashor's Tooth

Rabadon's Deathcap. Again, not up for debate. 120 AP + 30% Bonus AP. Seeing as AP helps Teemo's basic attacks, Blinding Dart Noxious Trap deal more damage, its a must have item. To build this item, first buy Needlessly Large Rod which will give you a fast and high amount of AP. It costs 1600 gold. From there you can do two things: Wait for 1700 gold, and just finish the full item, or buy Blasting Wand THEN finish the full item. Either way.

Next up is Runaan's Hurricane. I feel that this item was made just for Teemo.It makes your basic attacks hit 2 additional targets, though at 50% damage, but it applies all on-hit effects. Think of this. Teemos basic attacks, with full build, will be doing about 800 damage each hit, plus Toxic Shot they also deal DoT. While the basic attack itself is reduced by 50% damage for the two other targets, it doesnt cut the damage from Toxic Shot or the DoT from it. Add to it that it gives you 70% ASP, it makes Teemo great with farming, single target, and even better in team-fights because he will be hitting three champions at the same time, and poisoning them as well...No particular buying order is required for this. Just make sure to buy the whole item, obviously.

Now for Blade of the Ruined King... 25 AD, 40% AS, 15% LS. That alone makes it a great item for Teemo. Especially if you are coupling it with Runaan's Hurricane... you are hitting three targets, and getting 15% LS from each of them.
Then, it does 5% of current health of the target. Not the total health, but the current health. So they have 1000 hp, it is dealing 50 Bonus damage. This is great for champions like Volibear that can end up with about 4000 hp at the end of the match. It helps you burn them down as fast as possible. Then again, you are hitting three targets for that same amount since Runaan's Hurricane doesnt lower the damage from other items, just the base damage from your basic attacks.
Then the next part of it. You can activate it to deal 15% of the targets MAXIMUM health. Not current health. This active ability is the opposite of the items passive. So Volibear has 4000 health, activating this will deal 600 damage right at the start. IT will also heal you for that 600 damage, and steals 30% of their movement speed. At this moment I am unsure whether it gives you that 30% stolen movement speed, or if it just slows them. I will update this when I can figure that out.
To build this item, Just save up to get Bilgewater Cutlass then the full item.

Next up is Void Staff. As a note, the linked item shows 35% magic penetration. The actual item says "Magic Damage ignores 35% of the targets Magic Resistance, applies before Magic Penetration". This means that you IGNORE 35% of the targets Magic Resistance (MR). It may sound like it is the same thing, but its not. Let us say you have an item that gives you...25% magic penetration. That means you are "breaking through" their magic resistance...
Void Staff on the other hand, pretends that they dont even have that 35%. Meaning that if you have Void Staff and an item with 25% magic penetration... They have 100 MR. Void Staff lets you pretend they only have 65 MR left. Then your other item, 25% magic penetration, ignores 25% of that 65 MR, further reducing their MR to 48.72.

The next to items are situational.
Wit's End and Phantom Dancer.

Wit's End gives you 42% AS, 25 MR, and basic attacks deal 42 bonus magic damage.
The fun part of it, is that it STEALS 5 MR from the target on each basic attack, giving you an equal amount of MR. Meaning it lowers their MR by 25, which helps you deal more damage, and in return you get 25 EXTRA MR letting you take less magic damage. Totaled out, with maximum stacks from this item, it increases your MR by 50. So this item is great when fighting at mid lane against a strong AP Champion, or if you want it gives you an advantage over most AD champions that usually have an already low amount of MR.

Phantom Dancer is also a great item for Teemo. 50% ASP, 30% Critical chance, and 5% movement speed. The critical chance applies to your AD and AP damage. But to me, the most important part of it... is that it allows you to move THROUGH units. Minions, monsters and Champions are now nothing more than air when you need to move. I can't even count how many times I have tried to run from a gank... and then met a solid wall of my own minions blocking my path, getting me killed. Also on the flip-side of that coin, it allows you to go through all minions when chasing a champion.

The order you pick your skills for Teemo is entirely situational, and as such can be broken down into a few different scenarios.

Lets say you are top lane. Against...Pantheon. His passive will block any incoming damage one time for every four basic attacks Pantheon makes. He will have a circle of Red Shields surrounding him when this ability is up. If you hit him with Blinding Dart he will block the initial shot, and the blind effect. In order to counter that, hit him with a basic attack first. he will block the attack, BUT, he will still have the DoT from Toxic Shot. Useful to know that. But then you can follow up with Blinding Dart making his basic attacks miss, which prohibit him from gaining another shield as fast as he normally would.
Keep in mind though that while Blinding Dart will make his basic attacks miss, he can still hit you with his activated abilities.
So when fighting somebody like Pantheon it is best to put your first point into Toxic Shot and then into Blinding Dart, then alternate until you have two points in each. Then put one point at level 5, into Move Quick, then into your Noxious Trap at each available level. Or against a burst champion like Xin Zhao, you blind first, then auto-attack. Blinding a champion like Xin Zhao can make him waste two of his abilities because they cause extra damage with his basic attacks, which are now missing because he is blinded.

But lets say you are at mid. You are fighting Cassiopeia. Two of her abilities will target the ground beneath you, and strike in a small radius. In that case put a point in Move Quick first, and then into Toxic Shot and alternate until level 5, then put a point into Blinding Dart, and a point into Noxious Trap at level 6. Against a champion like Cassiopeia, Blinding her will only make her auto attack miss. Since Cassiopeia is a pure AP champion, it is pointless to have her basic attacks miss since it is her activated abilities you want to avoid. At that point Blinding Dart just becomes a harassing damage dealing ability.

As my own personal opinion, I dont think Teemo should ever be support of ADC. Not support because he deals too much damage. Not ADC because he just isnt an ADC. Also I recommend not going jungle. it is possible. But its just not practical for Teemo. Although I would recommend "Wandering" across the map to do some ganks. You pasivly move faster than most champs do even when they have boots, you can blind, then burst damage very easily. And since you are faster, it is very hard for them to escape you most times.

The best advice I can give on HOW to play Teemo.
Try to be sneaky, even when soloing a lane. Teemo's passive makes him invisible. Take advantage of your opponents limited vision. Stand in the middle of the lane when they arent looking, turn invisible, and wait.
Pop out of stealth with a mushroom right under them, or to block their retreat, hit them with Blinding Dart then follow with your basic attacks until they die. And if you have Ignite, use it at this point. And if they somehow make it away with a little bit of hp, Use Flash if you have it, and get in some more shots.

My personal way... I make sure that I always have at least one Mushroom from Noxious Trap at all times. This will help against surprise attacks, or if you are fighting a melee champion. It will punish them for trying to close in on you. Also once you have your Noxious Trap make sure to use them everywhere. BUT, try to avoid puting them in the middle of the lane unless you know it will hit a champion. Nothing more annoying than putting a mushroom down only to have it hit two minions and do nothing.
Make sure to put your mushrooms in places like the bushes, or around the corner of a place you cant see. Its a bit harder if you are mid lane because you have to cover three points of access at all times. Also, make sure to use the activated abilities of any items you have.