This build will allow you to turn Teemo into an ultimate poison-dart-firing machine!

Marks
For Teemo, the best Marks to get are the Greater Marks of Alacrity, which are Tier 3 attack speed Runes that give you +1.7% attack speed per rune. Some basic math will tell you that 1.7% x 9 = 15.3%, which will help you a lot early-game and mid-game. By late-game, your build should be complete or almost complete, meaning you'll already have around 2.3 attack speed.

Seals
Teemo already has a lot of attack speed with this build, but having a lot of extra attack speed early-game is crucial. The best seals are the Greater Seals of Alacrity, which give +1.7% attack speed each, which in total adds up to 0.76% x 9 = 6.84% attack speed.

Glyphs
Teemo will need a bit of magic penetration, so the Greater Glyph or Insight should cover that, since they give +0.57 magic penetration, which totals up to 0.57 x 9 = 5.13 magic penetration.

Quintessence
The Greater Quintessence of Insight is also great for AD Teemo. Each of these will give you 1.89 magic penetration, which added to the 5.13 magic resist from the glyphs totals to 1.89 x 3 = 5.67, then added to 5.13 = 10.80 magic penetration.

The best skills to use for Teemo are Flash and Ignite. If you don't like one (or both) of those spells, there are other options that are good for Teemo, such as Exhaust, which will make it easier to secure a kill on the occasional opponent who is faster than you. There is also

BEST SPELLS
Exhaust - Exhaust is great for chasing on Teemo, but the only down-side is that it's probably not as good as Ghost.

Flash - Flash is good on almost any champion, even the ones who already have abilities that do similar things to Flash, like Shaco. Teemo is very squishy for most of the early-game, so you'll be one of the main targets (unless you've got another squishy teammate), so you'll be running a lot, and Flash is very useful to have when you're escaping.

Ghost - Ghost is the best spell for chasing and escaping. I normally don't use it as Teemo, because he's normally quite swift anyways, but it is a very good choice.

Ignite - Ignite is a great way to get a kill on an escaping enemy when they're at very low health, but should not be used against the Dragon on Treeline or Baron Nashor on Summoner's Rift.

GOOD SPELLS

Clarity - Clarity is helpful for recovering your mana and helping you lane longer, because Teemo needs to keep placing mushrooms. Unfortunately, Teemo's health will be the main reason he recalls a lot, so you'll be refilling your mana at the Summoner's Platform a lot anyways.

Cleanse - Cleanse is great for removing crowd-control effects, but the chances are if you've got enough crowd-control effects on you early-game, you're probably going to die anyways and they're just trying to secure the kill even further. Late-game this could be useful, once you have Frozen Mallet and start getting a bit tanky.

Heal - Heal would be useful for Teemo early-game, since it could potentially save him, but you'll be getting enough health potions early-game, so this would be rendered useless most of the time.

Teleport - Teleport is good on Teemo because it also enables you to teleport to Teemo's mushrooms, which can prove useful for ganking and saving turrets.

WORST SPELLS

Clairvoyance - Clairvoyance is good for lots of characters, but Teemo is better of with spells that help him chase and survive.

Promote - Promote can be useful, but not for Teemo. It's used to make your team's minions stronger, which is helpful for pushing, but is unnecessary for Teemo.

Revive - Revive allows you to revive instantly after dying every few minutes. It's not really useful on Teemo (or most champions, for that matter), unless they're in your base and are attacking your Nexus.

Smite - Unless you're planning to jungle with Teemo (despite him not being an amazing Jungler), do not get Smite. Smite is useless if you are laning, especially on 3v3.

Surge - Not a lot of people use Surge anyways, and it's actually quite under-rated. However, it isn't great on Teemo. It increases your Attack Speed by 35%, which you won't really need since most of the items Teemo will use increase by more than 35%.

You'll want to participate in team-fights as much as possible, as Teemo's poison is a great way to finish off enemies. You'll want to either be behind the tanks firing at the enemies, or hiding in a brush where the enemy will be retreating to. It's a good idea to place mushrooms in the areas where the enemy will be running to, so that you can slow them and move in for the kill.

The best part about Teemo's poison is that if they manage to escape with a tiny bit of health, the poison will finish them off, unless they've got health potions, Summoner Heal or a good Support Champion on their team.

Teemo's poison does make it easy for him to KS (Kill-Steal), so you should be able to get most of the kills in a team-fight. This doesn't mean you should abuse this and not let anybody else on your team get fed, so don't go out of your way to KS a teammate who obviously has the kill.

One of the most important elements of any champion is their ultimate, and Teemo's ultimate is probably one of the best for helping your team. Teemo's ultimate allows him to place a poison-mushroom (cool-downs do apply, and he can only hold up to three at a time) that explodes and, like Teemo's regular poison, does damage for a few seconds after the hit. These mushrooms also slow the enemy, making it easy for a nearby champion to secure the kill on a running enemy. You don't want to save up mushrooms and use them all at once in the same place, because they don't stack like that. Instead, place them as you move around the map in places that will help your team out (similar to warding), so that they know if an enemy is coming, and in places where you think enemies will run past or try to recall at. You should never have three mushrooms stacked at once unless you're out of mana and are tearing down the enemy Nexus.

Pros

Lots of attack speed will help you secure a lot of kills
Blinding Dart can enable you to take out most champions who have more health than you if they're AD
Swift

Teemo's passive (or "unique skill") is his ability to camouflage after standing still for two seconds. This makes it very easy to gank them or ambush them from the brush, because their whole team can be standing on top of you and not notice you until it's too late (that doesn't mean you should engage them all just because you've caught them by surprise, but it does make it easy to ambush them, gank and hide.

The only down-side to this is that if they have an Oracle's Extract or a Vision Ward up, you're kind of in a tough spot.

Farming as Teemo is exceptionally easy. AD Teemo will typically have a lot of attack speed, so you'll be able to get most of the minions in a wave without having to try and last-hit (though that is an easier way), since your poison will finish off a lot of minions and you'll be hitting so fast that your team's minions won't be able to steal your minion kills before you.

Teemo's mushrooms are a life-saver, because if you or one of your teammates are running away and are being chased, if you run through one of your mushrooms, they'll probably chase you into it and get slowed and take some damage. This can even turn a fight around and allow you to get a kill.

They're also good for saving characters like Jax and Lee Sin, who can use their skills to jump/dash to Teemo's mushrooms, even through walls.