As AP Kog'Maw, your job is to poke down the squishiest target from as far back as possible. Due to Presence of Mind, if you get a takedown all of your mana for the next 5 seconds is completely refunded. So don't worry about stacking your ult too high, just keep poking until you lose the effect.

• In lane, use your E (Void Ooze) to poke down your opponent. This works well with Arcane Comet as the slow makes it almost un-dodgeable.
• You can then follow up with a quick Q (Caustic Spittle) while they're still slowed to deal a decent amount of damage as well as lower their resistances.
• Pop your W (Bio-Arcane Barrage) and basic them a few times to whittle their hp down even further.
• Finish them off with your R (Living Artillery). Coup de Grace works in tandem with your ultimate's execute damage to punish low health targets even more.

Remember, Presence of Mind also procs after leveling up, so use as many abilities as you can after leveling to maximize free damage.

Max R whenever you can cuz why would you not lol.
Maxing E first makes it deal LOTS of damage as well as slowing them for a pretty good amount. Unlike your Q, it goes through minions allowing you to land it more reliably in lane.
Max Q second for the high damage skill shot and armor/mr shred is nice too.
Max W last because it's not that good for AP kog and you probably won't be close enough to land those autos anyways.

Starting Doran's Ring instead of building straight into Tear of the Goddess is a game-by-game decision, and really depends on which one you think you'll need more. The Doran's ring is nice because it makes Absolute Focus give you AP instead of AD, otherwise you would have 0 AP from items and the Adaptive Damage would be turned into AD which is kinda useless for you.

On the other hand, building straight into Tear from the start gives you more mana to work with and allows you to scale slightly quicker, so once again it just depends on the game.

Everything else is pretty much set in stone: you need Archangel's Staff for the mana, AP, and mana refund passive. Also it gives you more AP the more mana you have which give you a pretty big powerspike when you buy Morellonomicon.

Rylai's for the slow on your ult, making it really easy to follow up with more ults.

Morellonomicon for more AP, mana, and the mana refund after takedowns gives you a buffer if you can't kill someone in 5 seconds with Presence of Mind.

Luden's Echo and Rabadon's for the most AP possible. Once you're full build with elixir, depending on the time in the game (due to Gathering Storm) you will have somewhere from 800-1100 AP, meaning you can kill ADC's in about 3-4 consecutive ults alone. Then you win.