Not Updated For Current Season
You've probably played Teemo with various styles, this is a hint how I play him.
I think Teemo's most funny (and annoying for enemies) ability is Toxic Shot, now that it has an impact damage, you can take the most damage from AS and AP. So I focus on these stats.
Teemo is a fast leveler so I don't use flat runes. You can use whatever rune you want, but i highly recommend the mp5 seals 'cause Teemo is real mana hunger in the endgame because of his shrooms.
I tried many summoner spells, but this combo is the best so far. I recommend using Teleport as an "ambush" spell: port to your Shrooms thats behind a wounded enemy, it'll be a kill most of the time. Flash is an escape mechanism, but you can use it to chase enemies, an hit them with a Toxic Shot.
You can go in whatever lane you like, it's real no big deal. But if you are not alone, try to work with your mate in your ambush-ports or just pop out from the brush and help him, when it's the least expected from the enemies. At first I buy Meki's, then as fast as i could Mejai's. Tried to start with Mejai's but it was no good so far (don't know why...). Then Malady, the rest sequence is up to you. (But I suggest to use the items upwards.)
This type of Teemo is a real badass in the mid game. When you have Nashor's and Berserker's, you have about 1.9 AS (2+ with Guinsoo's), its real great considering the low price. If you just harass enemies with Toxic Shot, and let them walk in your Shrooms, then take a finishing blow, Mejai's will build up nicely.
Teemo is not a perfect end-gamer, you are quite fragile, low health, low defense skills. That's why I use the Elixirs, they are perfect for you. But don't be afraid, you are one of the fastest champions. Try to pop out, make a shot, then retreat, and try to position yourself to make enemies to explode your Shrooms. If they get caught, then you have the least to fear, probably they'll run. :P
Put your Shrooms in the brush and in the corners (where enemies are likely to move), where enemies make escapes and don't be afraid to use them.
Try to port to your Shrooms behind enemy lines, and attack them from the back.
Think before you act, don't rush in a fight, without a set up of Shrooms, or a ready escape plan.
Switch Move Fast when you are up against a melee, and try to stay out of range, if you don't want to attack.
Put Shrooms around your position to avoid ganks.
Blinding Shot is not a hard hitting ability, but could be a life saver, and a good harasser against physical champions.
I hope I could help, and please feel free to share your opinion about this build, or your style of Teemo.