This is a guide to how I play Teemo. I push for an AP focus but also with some AD/AS to give him a sort of hybrid edge. Pure AD Teemo in my mind is a complete waste for his abilities as they definitely lend you an edge in combat.

Pros:
Mushrooms allow for great map awareness and give rise to many ganking opportunities.
Blind is good against AD attackers
Good movement

Cons:
Very squishy
Does not have a lot of weapons at his disposal. Just Q and auto attacks.

The runes I use allow for an early bang. AP quints and glyphs will give you 24 AP starting and lets your Q do some nice damage even at level 1. There is absolutely nothing wrong with having more AP. I take magic penetration marks since this build focuses primarily on AP. For seals I use dodge seals for some slight defense.

Other viable options:
-Seals: Armor/level, flat HP, mana/level (though once I get a second dorans ring I never run out of mana)
-Quints: Magic penetration quints are also very viable. Penetration is godly.

Ignite and Exhaust are best for murdering your enemies. Other viable options are Teleport, which is great for TPing to shrooms to get around the map as well as Ghost and Flash for getting in or out of trouble. I don't like Ghost since Teemo has his W active but it is certainly an option for super zoom zoom speed.

Spells I would not take include:
Heal: This spell is for noobs
Revive: You should not be dying to the point where this could ever be useful.
Clarity: The only time you might have mana troubles is around level 6 when you start placing mushrooms all over. This is not an issue once you pick up a second doran's ring
Rally: Does anyone use this ever?
Cleanse: Nah.
Fortify: Nope.
CV: Your mushrooms if placed well give plenty of map vision.

Masteries are set with lots of offensive focus. This build is designed to murder your opponents, not to support or just play with mushrooms. Basically a point for both summoner spells, and a point in everything else that could potentially give you a little extra killing power. The leftover points I throw into armor and magic resist.

R > Q > E > W.
I take a skill in W at 2 or 4 for escape situations. Finish Q then E taking points in R when available for maximum damage. Once your AP starts to build your Q will be hitting like a truck.

Dorans ring. Dorans are great early game. I start with one and build 2 or 3 of them. With the AP runes your Q and E will be hurting from lvl 1. After a few rings, get your sorc shoes for more speed and magic pen, then grab a nashors for some all around goodness: AS, some CDR, more AP, mana regen? Mana isn't really a problem with stacked dorans rings. From there go ahead and grab a rabadons to kick your AP sky high. This is around the time you can start selling off those rings to make room for gear as needed. I grab a recurve bow for later build into madreds or rylais. Rylais will give you some HP to make you a little less squishy and the added slow + your shrooms should bring enemies to a grinding halt. This super snare opens up a lot of ganking to your team mates assuming you have good shroom placement. From there finish out madreds to give you some dmg and the passive goes lovely with your E. Top it off with a gunblade for some vamp/spell vamp, a snare trigger, and more dmg and AP.

You can go mid or to a side lane. Teemo is very flexible champion in terms of laning and his W will give you the option to run between lanes with some speed to assist. Start with your Doran's and just harass with Q. Your E will give your auto attacks some extra umph. While Teemo's passive allows him camo, do not try to be so obvious that you're hiding. Most people will instantly go towards that location to drop an ability on you, though you can use this for the haste bonus to drop in another Q and some extra auto attacks and send em back. I try to get enough gold saved up for AT LEAST another doran's ring, some boots of speed, and some health pots before I go back to base. Second time back you'll be wanting to finish your sorc shoes and either pick up a 3rd doran's or start on your nashor's depending on your cash flow. If cash is a problem I'd stick with doran's but if you're getting some kills and good farm then start on the nashors.

You'll want to finish up Nashor's and your Deathcap asap. Start lining the map with mushrooms to give you some map awareness in key bushes and jungle paths and start ganking. W passive gives you some good speed and allows you a getaway in case things go wrong. Unless an enemy is splattering themselves on your mushrooms, you shouldn't try to go after people 1v1. Teemo's greatest weakness is that he is very open to a beating once his Q is fired off...and that blind won't help you against Malzahar's and Lux's and what not, so make sure to gank with your team. Don't be afraid to get fed as your build is quite pricy.

By now you should have finished your deathcap and started just packing in the extra damage where needed through rylais and madreds. Blue buff is your friend for keeping mushrooms planted, as a well mined field can help you and your teammates get in and out effectively for push. The closer your build gets to finishing the more rape damage you put out. Recurve bow with nashors will give you decent enough attack speed to be pecking away at enemies between Q shots which they will abhor.