I am not iPav (the Challenger Teemo main on NA). I am just writing down the items and runes from his guides on Youtube, since I prefer to have everything in one spot to look at in game.

Here's iPav's guide for patch 7.23:
Play Video
Here's his update for patch 7.24:
Play Video
iPav is a very informative Twitch streamer at twitch.tv/ipav999. He also has a Youtube channel where he posts gameplay videos and guides like the one above. Watch his content to see in depth explanations, how he plays and what changes he makes to his builds.

There are two rune sets for this build. One vs tanks and bruisers that you want to auto a lot and one vs squishy champions and other champions you want to poke. The rune set listed at the top of the build is the one vs Bruisers and Tanks.

In that rune set, you can switch between Coup de Grace, Cut Down, or Last Stand depending on the matchup. Gathering Storm is also optional. You could use The Ultimate Hat, Manaflow Band, and occasionally Waterwalking or Nullifying Orb instead.

The Summon Aery rune page is on the second build order in the guide. They are the optimal runes for patch 7.24 with Summon Aery. (In case you can't load the second page for some reason: Summon Aery, Manaflow Band, Celerity, Scorch in the primary tree. Domination secondary with Taste of Blood and Zombie Ward.

One important thing to note is that the Domination tree and Taste of Blood is necessary because you have to take Dark Seal to get AP instead of AD from the Sorcery Tree. This means you don't have the sustain from something like Doran's Shield to rely on.

Start and max E and max W second vs Bruisers with Press the Attack

Start and max E first and Q second vs squishies you can auto attack frequently with Summon Aery. Q, then E vs champions you can't auto attack as easily (like Pantheon).

You probably want to watch the guide or iPav's other videos to see a better explanation of the different items. Warmog's in particular is an unusual item for Teemo, but he believes it is the best item for the combination of sustain and anti-burst on Teemo.

iPav has also expressed the opinion that full AP Teemo can work just fine in lower tiers of play, but he has to build defensively to consistently win vs players in high Elo, so that is another reason his build may look unusual compared to many standard Teemo builds.