I have seen many guides for AP kogmaws, I have seen some played ap kogmaws (most of people are playing popular AD build, not using his full potencial though). And after reading few of guides for AP kog (best rated ones!) I decided to show you how he should be played (I played kog with many, many builds and with this build, I am usually wining), I wont say AD kog is useless (its awesome too, ill look at other ad guides, if I wont find one that fits me, ill try to write one). And sorry, there wont be pictures :(

Pros:
[*] great farmer
[*] can pwn teams and dont even show near teamfight
[*] annoying spammer
[*] unbeatable 1v1 due to high dmg output in short time and pasive
[*] noone can run, you shoot ulti to fog of war

Cons:
[*] aiming with ulti is hard when laning against good players
[*] I wont lie, hes squishy, but after death, he can still do much harm in teamfight

NOTE !!you have to be mid from begining!!

There isnt "must have" combination of masteries, I am personaly used to teleport (old (but good) habit from dota, map control is power!) and ghost (mostly for running, but can be used to help lines/teammates catched in jungle/etc...)
- exhaust good choice too
- clarity mana hungry as he may be, he really dont need it (if you know when and how to use skills and have a bit of common sense)
- other summoner spells if it isnt rally and revive... still, i think first 3 ones are better

Masteries... Its posible to change Good Hands with Perserverence, I pick these two kinda randomly, you will use maybe good hands more than perseverence, you are hero who can score triple kill (I once saw even quadra) after death, but I am not encouraging you to die in every teamfight!

Runes: Pick quisences you have, all of these are working just fine

The other runes: you certainly dont need red ones (leave them to ad carries and chasers), and if you know when and how to use his skills, you dont need even blue ones (you have 40% cdr from items+masteries only). If he uses all of his mana and enemy team is alive, then ist something wrong, if enemy team dies, you have plenty time to regenerate by classic means

Core items are showed in the build, he needs magicPen boots (like I said, you have to be on mid at any cost, your enemy propably wont have more than basic 30+- MR, magicPen from boots will hurt him), he needs tear, (tear first, you should have one as soon as you got lv6, when you got money, blue pill to base, teleport again to line, enemy wont even notice and you can hapily farm your mana pool). Next item, build fiendish codex and upgrade to nashor tooth (or evil tome, but aspd + BAB + plenty of AP -> crying tanks). As any other ap carry, he needs rabadon and after rabadon, archangels (dont be afraid to use ulti for 40 mana any time, he can reach the tear cap really fast and in late game, it doubles his mana and he needs just mana pool).
After these core items, there are many options, be creative, if they have plenty of MR, void staff should be your choice, are they too fast? buy rylai...
Personaly (and i know its not good) I dislike defensive items on ap kog, he just have crazy range of ulti, slow, BAB which is granting aditional range of autoatacks... he shouldnt be in middle of fight. And if they focus you, you will die too quick even with defensive items.
And remember, after lv16, allways have elixir of briliance!

Many builds are suggesting void ooze and BAB, I have no idea why BAB, in begining, it only gives range, no big damage, you still have too low aspd to use it properly, and when they are after you, you need to do as much harm as posible in lowest posible time, BAB wont help you there, so caustic spittle is the right way.

I like AP build much more than AD build, cos I think... AD is not using kogmaws full potential, you are practicly using one skill only, but AP uses all of them (and they are all awesome)

Kogmaws bread and butter is his ulti... on 16th lvl, you got incredible range, rly annoying damage, ability to scout bushes for 40 mana with 0.6 sec cooldown, which is... when used well really OP, few remarks... You have to play few games with him to accuire good aim with his ulti and you should have bind to R key smartcast ulti (when you know its range, not in first game with him...), it really speeds up his dmg output (did I mention that you can kill any ranged carry in around 3 secs only with this skill?)

2nd most important, void ooze. with void ooze and ulti combo, you are able to kill whole wave of minions at 7th lvl (ooze to kill most of melee ones and ulti to finish ranged ones) it gives really nice aoe dmg with slow

caustic spittle is another nice ammout of dmg (but not aoe), due to its passive, it should be maxed before BAB, in teamfights, just use it on enemy tank, he will certainly feel that armor reduction

BAB... you may got impresion that I dont like this skill, but you are wrong, its just awesome... in lategame with this build, you got around 650+ AD, so its adding 6+7% of targets max health in magic attack... in combination with caustics pasive and nashors tooth... you just eat everything (HP mundos and chogaths are my favourite ones)

This build isnt for autoatackers... you are hard, game-winning carry, you have to farm, farm, ..., farm until at least lv 11, then you start to be usefull (and you grown to beauty)
I will add some informations later, this is rough skeleton, I am new here at mobafire and currently, I am sick, so excuse some typos (normally i dont have any problem with english or writing)
feel free to comment and debate, maybe try it few times before doing so, this build is by far best kogs AP build I have tried, fits me like glove made of my skin