Credits to ArrestedFoRseX for his build on AP/AS and helping me better my build.

Runes
With the great guide for runes here is how I chose them here. The guide also helps with other types of combinations. A very good read.

Spells
Teleport - Teemo's mobility is incredibly important, and given that you can Teleport to Mushroom locations, it's an invaluable skill.
Flash - Given that Teemo is the fastest (or one of the fastest) Champions, people love to focus him with slows and stuns -- especially early game, when he's really squishy. I find Flash amazing for escapes early and mid game. Late game, Flash is amazing combined with your speed for running down Champions and getting behind teams to drop Mushrooms in their path just before a team gank.

Core Items



Optional


Meki Pendant -- starting item
Health Potion x 2 -- starting item
Boots of Speed
Fiendish Codex or Malady -- depending on if you need the CD reduction or not. (Final)
Berserker's Greaves (Final)
Malady -- if not already taken (Final)
Sheen
Stinger
Lich Bane (Final)
Nashor's Tooth (Final)
(option 1:) Rylai's Scepter (Final)
If survivability and/or chasing is an issue.
(option 2:) Zhonya's Ring (Final)
If having no issues with survivability and just want more dmg.

Note: If I'm getting really harassed early game, I'll build a "Philosopher's Stone" over "Fiendish Codex". However, if I do that, I usually try to get the Codex later for the CD reduction.

A few other items to consider:
Guinsoo's Rageblade -- I haven't tried this yet, but the theory crafting makes it seem like it would be really good for a little boost to damage (mainly for tower pushing IMO), especially in combination with Nashor's Tooth. From what I hear, there's benefits to stacking two of these. Don't quote me on this, but something to do with it taking less hits to get a max AP boost from the passive.
Boots of Mobility -- The extra movement OOC would be awesome for getting behind groups to drop Mushrooms and placing Mushrooms near deep lane towers for later ports.
Mejai's Soulstealer - Gives AP and if you're able to stack then gives good CDR. Replace with Malady if you're owning early game you feel that you can handle stacks.


Attack Tactics
With an AP build, you're relying on Blinding Dart and Toxic Shot, along with your Mushroom placement for Champions kills. I consider this build more support and play it as such, but still manage a large amount of kills, while only a few death. Ideally you will rack up a ton of assist too.

The goal in on this is to bait them to you and in to your Mushroom(s) as well as your team. However, just simply off your tower or teammate long enough for reinforcements is fantastic.

Here's a few points on Attacking and retreating:
At all levels when solo against another Champion, I run in Blinding Dart, auto-attack and then retreat towards my nearest Mushroom (level 6+). This is usually easy to do without getting hit, however I don't turn on Quick Move until I'm retreating and out of range (or between their cool-downs), in case I do get hit, don't want it to be on CD with a faster Champ chasing or slow on me.
If they don't follow, or don't catch me, I wait for Blinding Dart to be up again and repeat number 1, coming from a different direction if possible.
If they start backing off, I'll push them until they are dead, at a tower or totally retreated.
Note: I've seen AP Teemo played aggressively well.