This is my personal Kog'maw build. It may not be right for everyone but it as worked for me quite well. This is an AD build for Kog'maw which, in my opinion, is the best way to build Kog. Kog'maw can be built either AP or AD.

This is my first build and I am planning on adding on and getting more descriptive as I continue to play as Kog'maw. I will start to take screenshots and also taking videos so you can see how I play Kog'maw.

I will be updating this frequently. Please give me feedback, good and bad! I will try to answer any questions and also include more detail.

Pros

High Damage Output
Large Range
Carry

Cons

Squishy
Targeted in team fights

I choose to take Ghost and Blink because Kog is squishy and is targeted in team fights once they see the high damage output. Ghost and blink can also be used to catch up to enemies that are running away. Another option to take is exhaust to slow enemies.

Kog'maw has a simple rune set up. All of his abilities are based magic based so magic penetration marks are used. Both seals and glyphs are attack speed. For quintessesnces, either magic pen or attack speed can be taken depending on what you would like more of. I choose to take attack speed so I am able to kill the enemy more swiftly.

This may not be the most efficient way to set up masteries for Kog but it has worked for me. A strong offensive build ensures more attack speed and critical strike chance. 5 points in utility help with a small amount of health and mana regeneration. Haste ensures you have enough speed to catch up to enemies.

Items for Kog are simple. The more AD you have the faster you will kill the enemy. I start off with dorans dagger because it gives me mroe AD and also a small amount of lifesteal. I have tried to start off with boots but have found that it works better to start with the dagger. mardreds is a must when using Kog. It will always help against tanky dps which seems to be the in thing to be lately. Zeal and Phantom Dancer give Kog the extra movement speed, attack speed, and the critical strike chance to bring down enemy champions. Infinity Edge is for the AD and the critical strike. Once I have boots, phantom, madreds, and infinity edge it is usually game over. The last 2 items are up to you. I like to get starks so I have lifesteal and attack speed. Trinity force is good for the slow so you can ensure a kill

Since attack speed runes are already in place I like to get Bio-arcane barrage first so I can hit from a distance. Caustic spittle is next for a boost of attack speed and the void ooze to slow the enemy. Maxing Bio-arcane barrage is the best way to play because you are able to hit from so far away. Void ooze doesn't do much damage since this is an AD build so that is maxed in late game. You only need it to slow your enemy. Bio-Arcane barrage will be the skill that you will use the most other than Living Artillery. Living Artillery takes some getting used to. When chasing down an enemy don't forget to aim ahead of them so that they run into it. I have found that when an enemy is stationary it is better to aim behind them because when they see it, it is instinct to run back, and you will hit them! People are very bad at dodging Kog's ult.