3/3 Argmage's Savvy (for some additional damage with your spells, though you are not AP)
2/2 Utility Mastery
1/1 Blink of an Eye
1/1 Presence of the Master
I chose a 8/0/22 build because the way attack speed functions you do not need to spend the extra points in the offense tree to get the 5% damage, and other benefits, but going down far enough for the +4% attack speed is good. Putting your other 22 points into the utility tree gives you some extra gold generation which will help counterbalance not having high AoE damage and you gain 3% movement speed which helps you kill and avoid getting killed. At the bottom is 15% reduction in cooldowns for summoner spells which is very handy as Kog'Maw considering your only escape mechanisms are Ghost and Flash (which you should have taken the improved versions of, giving you an already lowered Flash CD and improved Ghost). Additionally, in the utility tree you will find +30% duration on monster buffs. Since you are not ability power you will have low mana regen so the golem buff, along with giving you cooldown reduction, will give you almost infinite mana to slow and slay your enemies. The lizard buff is key though, it gives you an additional bit of damage to every hit which will be massive when you are hitting two times per second AND a slow which is amazing before you get Frozen Mallet.
I went with health seals and quintessences to improve your survivability late game because until you get your frozen mallet you will be targeted as one of the more squishy characters on your team. For marks and glyphs I chose attack speed because, especially early in the game, having that bonus attack speed will help you kill towers and destroy enemies while you have your Bio-Arcane Barrage active.
I go full mobility because Kog'Maw has almost no way to escape enemies, and no way to catch up if you have almost killed them and your Bio-Arcane Barrage has run out. Flash and Ghost accomplish almost any movement needs you have, whether it is flashing over jungle walls to escape hordes of enemies or ghosting to catch up to Ashe as she runs away with a delicious single bar of health left.
1 - Bio-Arcane Barrage
2 - Void Ooze
3 - Bio-Arcane Barrage
4 - Caustic Spittle
5 - Bio-Arcane Barrage
6 - Living Artillery
7 - Bio-Arcane Barrage
8 - Caustic Spittle
9 - Bio-Arcane Barrage
10 - Caustic Spittle
11 - Living Artillery
12 - Caustic Spittle
13 - Caustic Spittle
14 - Void Ooze
15 - Void Ooze
16 - Living Artillery
17 - Void Ooze
18 - Void Ooze
Upon death your character becomes invulnerable for four seconds and after those four seconds you explode and deal 100+(25*Lvl) damage to nearby enemies (both creeps and champions).
Don't rely on this ability for kills, but it is really nice if you almost kill someone and happen to be able to run up and explode them. Sometimes they even get really pissed and /all about how it needs to be nerfed. :P
Passive: Increases Attack Speed by 10/15/20/25/30%
Active: Deals 60/110/160/210/260 +(70% Ability Power) damage to the target as well as reducing their armor by 5/10/15/20/25.
I use this only if I really need to smash whoever I am killing, the bonus 260+ damage at level 18 is nice along with the minor amount of bonus damage from the armor reduction. The range isn't super long so don't try to use it too often, you are usually better off just decimating people from far away with your Bio-Arcane Barrage.
On use causes your attacks to have an increased 140/180/220/260/300 unit range as well as making your attacks do an additional 2/3/4/5/6 (+1% of your AP)% of the target's health in damage. This effect lasts for six seconds.
This is your primary ability. Use this to both harass and scrape enemies in battles. Paired with your Madred's Bloodrazor, Wit's End, lizard buff and Sword of the Divine you will be doing absurd amounts of damage on each hit, at an extremely long range. Because of its low mana cost you can use Bio-Arcane Barrage to harass early game as well, and potentially even get off some early kills.
Sprays ooze in front of you at a fairly long range which deals 60/110/160/210/260+(70% AP) to any enemies it passes through. Additionally it slows enemies who pass through the trail left behind by 28/36/44/52/60% of their movement speed.
This ability is great paired with your Bio-Arcane Barrage and a teammate's stun. Having your enemies slowed along with your increased range allows you to stand still and attack with no movement in between for a few seconds while your enemy is trying to flee.
Ultimate: Living Artillery [R]
You fire an artillery ooze at an incredible range (1400/1700/2000) which falls after a short delay dealing 80/120/160 + (30% AP) magic damage and reveals targets for 4 seconds. Damage to champions is increased by 150%. Every subsequent cast within 6 seconds costs an additional 50 mana, up to a maximum of 250 additional mana.
This is a great ability for killing fleeing foes who have managed to escape your spit storm, revealing potentially trapped bushes and farming minions when you need a bit more area damage. Do not use this as your main source of damage... it is good harassment, but it comes at a high cost if you are spamming it.
Start off with a pair of Boots and two health potions. Having some bonus movement speed will help you avoid death in the early game.
Now you get into the real items. Begin with a Recurve Bow to start building your Madred's Bloodrazor. Try to stay in lane until you have enough, but if you have to go back before you can complete your Madred's then go ahead and pick up either of the two other components. Madred's Bloodrazor is key to your face smashing. Combined with the lizard buff and Bio-Arcane Barrage you will be melting through players' health.
Now finish your boots with either Sorcerer's Shoes or Mercury Treads in the case that there are lots of stuns or other CC mechanics being used by the other team. There is no need to get Berserker Greaves anymore because of the buff to Caustic Spittle.
Next I go for Wit's End in most cases. If there are lots of dodge type characters or if there are more than one characters with no mana on the other team go ahead and get your Sword of the Divine instead. Then get Frozen Mallet after your Phage. Boom extra health and a slow so you don't need lizard buff as much to kill people.
After your Frozen Mallet it is a bit of a toss-up. I have been experimenting with Atma's Impaler and Warmog's or Guardian Angel or Banshee's Veil. I think that going Guardian with Atma's is the best combination since you tend to get focused after they figure out that you are the one nuking their faces off. Atma's Impaler will give you an extra 60 damage or so against all character on every attack once you have level 18 with your Frozen Mallet (3k approx. HP). If you use Atma's Impaler and Sword of the Divine you can just use an elixir to attack speed cap yourself if it gets that far into the game... If you stay with Wit's End (which in some cases will still be a better choice because of its higher damage) you will get almost to attack speed cap (which is plenty).
Try to lane not in the middle. While Kog'Maw is great middle, it is more useful for tank spanking on one of the side lanes. It makes a great combination with almost any characters. Pairing with ranged who have slows or stuns is a lot of fun, generally allowing some very early game kills.
If you end up opposing healers do not fret! You will out-damage almost any healer very early on in the game if you and your partner focus fire their dps. With your ridiculously large range and slowing effect and your partner's hopefully not too bad damage you should be able to bring down anyone in a matter of seconds.
Generally I play Kog'Maw very offensively, using Bio-Arcane Barrage to harass every time I can. It is very effective to combine your Void Ooze with your Bio-Arcane Barrage. Your enemies will most likely stand there and hit you for a moment before deciding that they cannot actually out damage you and your partner, at which point they will try to escape... but no, they have been slowed and are about to have their corpse feasted on. However, make sure you do not over-extend yourself. Even with Ghost and Flash Kog'Maw can be very squishy early game (however, once you are dead, you can always go blow them up!).
Now that you are level six with some good attack speed you should be able to harass very well with your ultimate, Living Artillery. Make sure that you do not use all of your mana for it because when you have your enemy at half health you will want to Bio-Arcane Barrage and Void Ooze them into a pulp. This is the major set-back of this build on Kog'Maw is his early game mana consumption. Eventually you will want to get the golem buff which will give you enough mana regeneration to spam your Living Artillery all day.
As you approach level 9-12 you will probably start grouping with your team and pushing lanes as a group of 3-5. Your job is to stand back, slow, and omgdestroy tanks (or DPS if there are no tanks there). Use Caustic Spittle if you are group fighting with a physical damage player as it will improve their damage output, if not then don't bother getting close to enemies because you are squishy!
Late in the game you will be grouped with at least four of your teammates which will hopefully give you some buffer and allow you to stand still and nuke players. Use your Living Artillery to check bushes, deal damage before fights start, and to kill fleeing characters. When fights do break-out go for the highest health players in order to demoralize their team (as they see their tank transformed into nothing but goo in a matter of seconds they will not be very interested in coming any closer) and to, of course, destroy their tank.
Remember that many people under-estimate the range on your Bio-Arcane Barrage so you can rush in, void ooze, and pop it to decimate their health. Rarely will they have time to run away.
Ghost and Flash will become increasingly crucial in late game as the damage intake will become higher and higher. Using your Flash well to escape death and your ghost to get away as well will make or break your game.
Neutral Monster Buffs
As previously mentioned, you should be grabbing the neutral monster buffs whenever you can. By level six or seven, possibly eight or nine if you are having a slow game, you should have enough power to drop the golem or lizard without taking substantial damage. These buffs, especially the lizard buff, are going to vastly improve your ability to feast on the tears of your foes... and the rest of them as well.