On-hit Teemo is extremely strong. Here's my guide to being one of the most annoying champions in the game to lane against.

EDIT: ADDED AP TEEMO BUILD

Start Boots of Speed + 3 Health Potions OR Cloth Armor + 5 Health Potions

You should then aim to buy Phage, Wriggles (if vs AD lane) or 1-2 dorans blades (if vs AP lane or dominating a bruiser) and finish your boots to Mercury Treads or Ninja Tabi depending on the enemy team composition or having trouble in your lane.

Now build a Wit's End as your main earlygame damage item. Finally, if you are getting fed, grab a Madred's Bloodrazor and finish Mallet next. If not, just finish your Frozen Mallet (perma slow) and slowly build up your madreds bloodrazor (main source of dmg late game).

At this point the game should be over. If not, go buy a Black Cleaver (need more dps) or ionian spark (pushing) or guardian angel (if fed) or something. Maybe double elixers and oracles, etc.

Early game you're going to need wards. Once you have Wriggles and your ult wards aren't necessary anymore. Put mushrooms all over the Baron area. It's also a good idea to place it in your lane and the bushes in your lane for bush control. Since you have mushrooms, you can even choose to place your Wriggle's Ward offensively or defensively for your jungler, such as in the enemy/friendly jungle. Make sure to be putting your mushrooms in frequently travelled chokepoints. It might net you a kill or two.

Since most likely you picked Teemo to counter an AD bruiser in the top lane, I took flat armor seals. MR scaling blues are always strong vs AP lanes and just in general. Reds and quints are up to you. I personally prefer using armor pen reds and ad quints although i think hybrid penetration runes would work just as well.

Flash is necessary on Teemo. Just an OP summoner spell.
Ignite and exhaust and teleport are all good choices. I prefer ignite because it stacks so well with your DOT.

These items might be popular on some builds, but really they're just a waste of money and space you could be spending on more valuable items.

Why not get Nashor's Tooth? It's too expensive for the ****py stats that it offers. You're not playing AP teemo anyway.

Why not get Malady? Wit's End is a better choice 100% of the time.

In teamfights, do as much damage as you can, and blind their AD carry if possible. If not, blind their bruiser and peel them off your own AD carry. Your bloodrazor allows you to shred just about anyone.

When split-pushing, notify your teammates so the enemy team doesnt secure an ace in a 4v5 fight. ALso, make sure there's no objectives up soon when split pushing. Finally, have good map control (shrooms and wards) so you don't get ganked. You can take the Demolitionist mastery if you're aiming for a quick tower kill at top lane.

-Don't use blind for harrassing, as it won't do much damage since you're not maxing it. Only use blind if the enemy laner has gotten close enough to you to attack you and seems to be somewhat commiting.
-Harrass, harrass, harrass! Push them out of lane. Be as annoying as you can. When you have phage you can go in for easy kills if they overextend.
-Teemo is AMAZING at kiting. This is improved with phage.
-Red Buff is great on Teemo btw. Try asking your jungler for it if they don't need it and you're purple side.

This build makes you fairly tanky and do decent amounts of damage. If you farmed well and played well you should be effective at all stages of the game and have the ability to score easy barons and objectives with your mushroom map control.

Notes:
-YOUR AUTOS HURT!! Auto harass attack as much as you can and blind as much as possible to push enemy laner out of the lane or net a kill. Even in mid Teemo is really annoying
-Much weaker due to dfg nerf
-Get your w at level 2 if you think you might get ganked early (lee sin, maokai) otherwise just get e
-weak in teamfights...low burst compared to mids but can pick off low hp targets
-can blind enemy adc...but will probably get cc and die before getting close
-if you can't kite through mushrooms, don't play ap teemo
-VERY SQUISHY, remember your zhonyas
-Recommend playing him mid but playing top works too
-provides even better objective control since mushrooms chunk hard

Build:
Core Items:
Sorceror shoes
Deathcap (get 1st or 2nd big item)

Situational Laning-midgame:
Abyssal scepter
Deathfire Grasp
Doran's rings
Chalice > Athene's (never actually tried this but with the cdr remove from dfg it might be necessary for shroom spam)
Haunting Guise (buy this only if you got abyssal but don't plan on buying void staff and didn't get dorans rings either. Very good for snowballing)

Situational late game:
Zhonya's Hourglass(great item if teamfighting)
Void Staff (must get if didnt buy abyssal and enemy stacking MR)
Lich bane (if you require more teamfight presence. Need higher burst and dps)
Guardian Angel (if you have no idea how to use zhonyas so get this instead)
Rylais (just for more hp, decent item but get it after 2 or 3 big items)
Nashor's tooth (Don't like this item personally but it's up to you)

Masteries - Go 21/0/9 or 21/9/0. I like the latter since im an aggresive mid
Runes - Magic pen reds and MR/lvl glyphs. Seals are personal preference and for quints i recommend movespeed or flat ap