This guide includes both of the builds that I believe are viable on teemo. I think that teemo is one of the stronger champions in this game, as he is extremely easy to snowball and very dominating in late game teamfights.

I suggest you always go with the On-Hit build, rather than the AP nuke build. The only scenario where AP nuke teemo is very viable, is if the match goes into extreme late game, or if your team is playing a late game comp; if that is the case, you definitely want to go AP Nuke teemo so your shrooms can win teamfights for your team, or even clean up when the enemy team is running away.

CC: Crowd Control (any movement impairing ability)
MR: Magic Resist
AP: Ability Power
MS: Movement Speed
GA: Guardian Angel
BV: Banshee's Veil
DFG: Deathfire Grasp
MPen: Magic Penetration
DPS: Damage per second
AA: Auto Attack

PROS

+ Very mobile Move Quick
+ Very good against dives Blinding Dart
+ Consistent Damage Toxic Shot
+ Free wards and huge annoyance Noxious Trap
+ Great Teamfight Presence (can blind enemy ad carry)
+ Great map control with Noxious Trap

CONS

+ Extremely low defensive stats (very squishy)
+ Very vulnerable to Oracle's Elixir and Vision Ward
+ Short AA range
+ Often focused

Seals: I grab Armor for an easier early game laning phase.

Marks: I grab MPen because it is the most important stat to be able to constantly output high DPS.

Glyphs: I grab AP to have a stronger on-hit bonus and more damage for my Q.

Quints: I grab MS to be able to run faster when ganking other lanes, to be able to come back to my lane faster after recalling and also to escape from ganks more efficiently.

If I am playing mid, vs. an AP carry, I will switch them up so I am able to get some MR instead of armor.

Build 1: Pretty standard teemo build, you might want to finish your Liandry's Torment before your Rabadon's Deathcap if the enemy team is building heavy Magic Resist. As for the 6th item, you can either chose to go for damage and CC ( Iceborn Gauntlet), Armor and MR ( Guardian Angel - which I would go for in most cases) or, if you feel you need more health, you can always go BV.

Build 2: VERY high AP is the goal of that build, maxing the damage of his Q and of his R, and adding the nuke potential with the Deathfire Grasp. As for the 6th item, two options are available: damage or defense. If your team is already ahead, I definitely recommend you grab a Liandry's Torment, as it will greatly increase his damage output.

Skill order 1 - On-Hit Teemo

Noxious Trap - Toxic Shot - Blinding Dart - Move Quick

Explanation -- Always get your Noxious Trap as soon as possible, so you can start placing mushrooms all over your lane. After that, you want to prioritize your Toxic Shot, since this is what the build focuses on. Then, max your Blinding Dart, so you can trade/burst effectively against your lane opponent. Lastly, get Move Quick, since having just a few points in it doesn't really matter.

Skill order 2 - AP Nuke Teemo

Noxious Trap - Blinding Dart - Move Quick - Toxic Shot

Explanation -- As I previously stated, always get your Noxious Trap as soon as possible, so you can start placing mushrooms all over your lane. Secondly, max your Blinding Dart, because it is the central point of this build (very high AP ratio on this ability). Then, I like to max Move Quick, so I am able to quickly escape after bursting my enemy with Blinding Dart. Toxic Shot should be the skill you max last, unless the enemy jungler is ignoring your lane, in that case, max your Toxic Shot before.