There's a lot of ways to play Teemo, but they mostly boil down to a few.
The most popular one is a hybrid Teemo, leaning more towards mushrooms than autoattacks and on-hit. That's the recommended items page.
My preferred playstyle is full AP: Landmine. With Landmine, you can be behind and still massively impact the game. One mushroom deals a surprising amount of damage. You can be across the map, splitpushing, and still win fights with the rest of your team with your mushrooms.
The other possibility is full on-hit. This would lean more towards traditional assassins, appearing out of nowhere(from stealth) to oneshot a squishy, and then make a run for it.

Mushrooms are what makes Teemo the demon lord he is. Put mushrooms at the edge of the river, and try not to let their trigger radiuses overlap. With four mushrooms, you can block the entire river opening.
One or two mushrooms at the entrances to the jungle can provide vision and discourage the enemy jungler from leaving their jungle. With one mushroom, place it at the center of the opening for the greatest chance to get stepped on.
Inside the jungle, put mushrooms around corners and on the outside edge of bushes.
A trick I like to use is to place mushrooms on the inside or outside of the base gates. Very few people will use control wards/sweepers in their base.
Spread your mushrooms out! Do not place all of them in one area. That's a waste, since mushroom damage does not stack. Now obviously if you had enough ability haste and mana to put down that many mushrooms, you can go ahead and turn Summoner's Rift into Iraq, but since not everyone has extreme amounts of ability haste, place your mushrooms at critical points.

Teemo is legendary for being annoying early. Poke your lane opponent into submission. Make them regret not banning Teemo. Make them regret playing top. Make them regret installing League of Legends. Make them regret being born. Teemo not only kills champions, he also breaks mentals while he's at it. If you can tilt your lane opponent early, they will make more mistakes that you can capitalize upon.
Early game is where Teemo shines: nobody can respond to his poke properly.
Your goal is not to get as far ahead as possible, although that is pretty good. Your goal is to put the enemy behind.
Tip: auto the enemy champion, then walk into bush to drop minion aggro. Do not stand in the bush when you are autoing or this will not work. Going into a warded bush also will not work.
Tip: play around enemy cooldowns. Once an enemy's key ability(Aatrox Q, Mundo Q, Yorick W) misses, you can go in. Having a general idea of their cooldowns will be very useful.

In the midgame, you are no longer the most cancerous thing out there. Start playing more safely. You can start considering going into the enemy jungle to lay mushrooms. If nobody comes to stop you, you can take towers.
Teemo's flanking ability isn't exactly great, but it's not bad either. Lay a mushroom or two across the enemy's retreat path, and then move forwards to try and burst down the enemy carry. Dropping mushrooms near objectives is a good idea.
Your goal is to push waves, and lay down mushrooms while you're at it.
Tip: you can teleport while stealthed, without breaking stealth. This is a good way to escape from an impossible situation.

Lategame has come. Everyone oneshots Teemo. In exchange, Teemo oneshots them!
Your main goal at this point in the game is to wander around the map, laying down mushrooms and pushing waves. Teemo is about as strong as a sheet of paper at this point, so avoid fighting anyone you can't burst down.
Your goal is to lay down mushrooms around objectives, and to push waves while you're at it.