Most Kog'Maws underestimate their true power with a full item set. Bio-Arcane Barrage should be used to help take down tanks or for the extra range in a team fight. When fighting opponents that are squishy e.g. Ashe, you can just right click and fire away.
Kog'Maw can out-range most champions with his Bio-Arcane Barrage ability. Use this to your advantage by harrassing enemies through the minion wave, or grab last-hits when being zoned.
Void Ooze is Kog'Maw's only escape ability. In order to succeed in running away, point Void Ooze away from your enemy towards where you are running. You will be able to run over it without a problem, but they will be forced to run on top of it the whole time resulting a drastic speed reduction. Ghost or Flash can give you the "lead" needed to compensate for the short casting time.
You can however, click on the map immediately after using Void Ooze to stop the animation, making the escape easier, though it can still be dangerous and ghost or flash can compensate when is really needed
Ability Power, combined with Living Artillery, can lead to devastating long range damage. This is a great way to prevent your enemies from camping by a turret.
If the Living Artillery hits a champion, your team will be able to see that champion for 4 seconds. This can be useful to continue auto attacking or using other targeted abilities after an enemy has fled into the bush.
It is worth noting that while these two items together increase damage output by a good amount, it also leaves Kog'Maw very fragile. Building resistances and a Frozen Mallet will allow you to stick on one target very effectively while being much less squishy.
With Bio-Arcane Barrage and Madred's Bloodrazor, Kog'Maw can tear down most high health champions. Because the effects are additive, it brings you to a total of 10% of your target's max health per attack, meaning with enough magic penetration/magic resist reduction, you can theoretically kill them in just 10 hits. Adding AP can boost this to 12% or more max health per attack.
Malady is a very good item for Kog'Maw, as he benefits greatly from both ability power and attack speed, and the passive magic damage and magic resistance reduction increases the damage of all of his abilities.
Wit's End is another early-game item you may want to consider for Kog'Maw, as it is a VERY efficient item for the price, and all stats on the item are useful to Kog'Maw.
Building AP or DPS depends on what the enemy team consists of. Tanky DPS and tank enemies will call for DPS, while a balanced or assassin team will need AP. This is because DPS will kill faster but expose yourself to danger, while AP lets you take off health from a reasonably safe distance.