Not Updated For Current Season
First, I've been playing Teemo quite a bit. I've got about 350 matches with him and I've played many, many builds. My preference is to play him with Ability Power, as opposed to Attack Speed.
My current build works well for me and I've been averaging about 14 kills per game. I know the Attack Speed build works for some people, but if I'm going to 'try' to nuke people, why not just 'nuke' people by playing Ryze, Eve, or some other awesome champion. Teemo is support, and as such, I play him in a support role.
I use my shrooms to help control the map. I can deter the entire enemy team from entering certain areas of the map, like near Baron. More experienced teams will buy elixirs and destroy Teemo's shrooms, but that's okay, because they could be spending money on more important things and doing more important things like killing your team. Killing Teemo's shrooms is a waste of time and if enemy teams choose to do it, well, that gives your team an advantage.
I have a VERY particular shrooming strategy. Location is everything. Understanding where people walk most often is key. Any place that you or a teammate can be ganked from should have a shroom in it. The entire team can see the shrooms as they provide a small area of visibility through the Fog of War. This is an excellent advantage that people don't always realize.
To maximize the effectiveness of Teemo's shrooms, not only do you need to place them correctly, they need to be potent when an enemy steps on them. Therefore, I recommend the Ability Power build, which adds damage to your shrooms.
Dying is the worst thing you can do, so I choose Heal + Ghost as my summoner spells. Heal helps me stay in lane early game + my health pot. Ghost helps me mid to end game and sometimes chasing enemies.