Hello, and welcome to my Kog'Maw build and guide!
I like to play Kog'Maw as a hybrid AS/AP carry. This is because his Caustic Spittle passive and a couple of key items allow him to easily hit the attack speed cap, while also dishing out amazing magic damage with a Madred's Bloodrazor and Bio-Arcane Barrage. With Bio-Arcane Barrage at rank 5 and active, he also has more range than any other character, not to mention his Living Artillery's range!
Disclaimer: This is how I play Kog'Maw. You can play him however you like, but I hope I can give you some tips/new ideas. You can change the build however you like, and if you feel like I've made a mistake, please leave a comment! I'm open to constructive criticism.

Runes, more than most other things in a build, are up to personal preference. Feel free to use whatever you find effective. As far as my runes go, I generally take the following:
Greater Mark of Magic Penetration - These are fairly standard for a build that deals mostly magic damage. Greater Mark of Desolation could also work, but you really are dealing more magic damage than physical damage.
Greater Seal of Scaling Mana Regeneration - Mana becomes a problem mid-late game, once you start spamming Living Artillery before a fight, and using Bio-Arcane Barrage whenever it's up. These help to mitigate that problem.
Greater Glyph of Scaling Cooldown Reduction - It's quite hard to get your cooldowns lower with this build, especially since you get Nashor's Tooth so late in the game. Also, there is most often a better candidate for the blue buff on your team.
Quintessences are where you have a lot more options. Greater Quintessence of Magic Penetration is pretty good, but so is Greater Quintessence of Movement Speed. You could even take flat health, if you wanted to.
Alternatives:
Greater Mark of Desolation
Greater Glyph of Scaling Ability Power
And more. Really, whatever you like.

I generally go 9/0/21 with my masteries. All you really need from the offense tree is the magic penetration, and the utility tree has masteries for Ghost and Flash. Be sure to pick up good hands, and not Perserverence, because those 6-7 seconds can be vital late game. Pick up masteries for your summoner spells of choice, and the cooldown reduction masteries from both trees.
You could also go 21/0/9, but only do this if you plan on building more physical damage heavy (AD, crit, etc) .

I always, ALWAYS take Ghost and Flash. The reason for this is simple... You are not Corki. You are not Ezreal. You are not LeBlanc. You cannot escape on your own once Xin Zhao or Master Yi is in your face.
Other possible choices (although I don't recommend leaving Ghost or Flash behind) :
Exhaust: This can also save you from the aforementioned melee DPS who wants the disgusting monster dead. It can also help get early kills off those who overextend.
Clarity: This is only useful if you plan on maxing Void Ooze first, and spamming it like crazy.

I usually max Bio-Arcane Barrage first, making sure to take a point in the others at level 2 and 4. This is because the extra damage and range help a lot in the laning phase for harassment. Laning against Caitlyn, and don't want to touch her with a 10-foot pole? You can even use this to hit creeps, if your laning opponent is that scary. After that, I work on maxing out Caustic Spittle. This is going to happen right around when you get your Madred's Bloodrazor, and the synergy between Caustic Spittle, Bio-Arcane Barrage, and Madred's Bloodrazor is beautiful. Be sure to take a point in Living Artillery whenever you can, the extra range is fantastic.

Start with Boots, and three Health Potion. These are good for poking and dodging early skillshots. Besides, you're way too slow. On your first trip back, get Berserker's Greaves, ( Boots of Swiftness are not a terrible idea either) and, if you've been farming real well or got first blood, a Recurve Bow. Why do I rush Madred's? Well, it makes your DPS EXPLODE. you can also get dragon easily, and Baron Nashor quite early. Be sure your jungler/roamer/support keeps these key points warded at all times. Malady comes next, because it synergizes well with the Bloodrazor/Caustic Spittle/Bio-Arcane Barrage combo. You will do incredible damage, even to tanks. Guinsoo's Rageblade is quite good, but if you want to skip it, it's not really as incredible as the other items, although the AP is very necessary at this point. Frozen Mallet, however, is amazing, especially since you'll pretty much be at a constant >2.0 attack speed if you got the Rageblade. The last item is pretty situational; here are a few options that I find viable.
Nashor's Tooth will basically let you keep Bio-Arcane Barrage up at all times, and the ability power is very good on you.
Banshee's Veil is great if they have a lot of CC, or you're getting bursted down by an AP carry ( LeBlanc, Annie, etc.))
Abyssal Mask is good if they're stacking magic resist against you.
Really you can take almost anything you want in this slot.

For starters, you're going to want to pick a lane. I recommend taking a side lane with someone bulky with a lot of CC, for example Blitzcrank, but if no-one else on your team wants it, and you feel confident, you can take mid easily. Because I focus on attack speed early, and don't start by maxing Void Ooze, farming is more difficult in mid lane, but it can be done. I don't recommend walking into the bushes, unless of course someone on your team was smart enough to take Clairvoyance, and you can know that it's safe. Last hit minions when you can, but don't just stand there auto-attacking them; this early in the game, you have little to gain by pushing, and it makes you much more vulnerable. When you get a chance, try to turn on Bio-Arcane Barrage and get a couple of hits on the squishiest enemy in your lane. Run away before you start to take minion aggro, however. Early kills are very good, but don't risk feeding the enemy any.

Mid game begins. Probably by now one tower has fallen, ganks are starting, and you're about level 7-8, unless you took mid. If you can push your lane, you can gank very well, just make sure that you aim Void Ooze behind the target, and ping the one you want to focus. Living Artillery is fantastic for getting runners, but you need to lead the enemy very precisely; it's quite easy to miss. Mid-late game, baron nashor is also a priority, make sure your team keeps it warded at all times. You can actually take baron fairly early with Madred's Bloodrazor and Bio-Arcane Barrage.

The most important thing to remember during a teamfight as Kog'Maw: STAY IN THE BACK!
I can't stress this enough. Stay behind your melee units, make sure there's a tank in front of you at all times. Your role is as arillery, you will never be in the thick of things if you can help it. Try to get Void Ooze down in the middle of the fight, it can be another helpful obstacle to block your enemies' path to you, if your tank dies/is not very good.

Basically, late game you want to keep farming, but always stick with your team if a teamfight is likely to break out. You can backdoor alright, but be sure to have map awareness; if an enemy goes MIA, it's time to GTFO. Try not to go past the outer tower; getting killed at this point really sucks for you and your team.