This build is how I play my ranked Kog'Maw. It can vary from item to item but I will talk about that more later. It is a slow and long build, built on creeping a ton and getting kills at start can help with the build going faster.
The greater mark of strength is the best choice of Mark because it helps with damage at start to get some poke off, as well as last hitting creep sooner than later.
The greater seal of resilience helps all game, but mainly at the start of the game. It helps to defend against other ADC's and supports to (even creep) survive in lane and remain creeping.
The greater glyph of alacrity adds on the attack speed that gives Kog'Maw his strength. It also helps add AS in the start with his Q to avoid having to buy an item of AS over AD.
The quint of desolation is huge for his endgame qualities. This gives him 10 armor pen which adds to his damage throughout the entire game. At the start it breaks bonus armor that the ADC or support has (I.E. taric passive armor from his armor skill)
The masteries I have chosen are for damage, speed, and survivability, considering Kog'Maw is a very squishy.
Resistance and Hardness (in the defense tree) are maxed out to help for survivability at the start of the game. The extra armor and MR are to help for bot lane creeping for the first 15 minutes which damage is low.
The rest of the offensive tree is basic talents for most AS/AD ADC. Few talents can be moved or changed depending on persons interests, but these talents have seemed to be useful for me for all ADC champs that I play.
The items and sequence I have chosen are very important. They help to harass early game once the BF sword is obtained and to give easy kills over people who build mass Doran's Blades.
Boots and potions are used to help get away or to the enemy champions, and potions are used for survivability.
The first BF sword is used to creep more and harass the enemy team which Kog'Maw's bio-arcane is activated. It may be hard but once this item is obtained the game begins to get much easier.
Full AS boots and life scepter are cheap and extremely useful to remain in lane and heal yourself.
Once available BT is a great first item because it has the most attack damage on any time once stacked fully. The life steal and attack damage should be obtained very quickly with the creep in your lane which still harassing the enemy ADC to keep them away from creeping.
The zeal will help give a small chance to crit, but mostly to give more movement speed to get to/away from enemies. Kog is a slow champion at natural and is hard to keep distance when under pressure.
Building the IE is an obvious choice of any ADC. This item should be fully built before the full PD because the attack speed built up with boots, max Q, and zeal is enough to be equal to or more than the other ADC easily. The crit damage is always important along with the damage on this item. Hands down, best ADC item.
Finishing off the PD is important next to get more mobility and AS.
One item I find important on Kog is Madred's Bloodrazor. This item adds more Life % damage to his W and as these two stack it can take out tanks and supports faster and easier. I always go for this item after the basic ADC items, but if the enemy team has large amounts of HP or many brusier/tanks on the team, I will go for Madred's sooner than fifth item.
The last item is always something situational depending on team composition. I chose FM on this build just because the slow can help Kog'Maw kite enemies allowing him to get away and kill at the same time, but also to help chase people who may take Ghost or have movement speed abilities (Garen, Eve, etc.). The health is also a big bonus because Kog'Maw is a very squishy ADC and is easy to get to without situating yourself right. Other items that are viable are basic ADC items, including; Black Cleaver, Extra BT, Banshee veil, GA, LW. All these items are dependent on the enemy team composition and how well your team is doing. Knowing the enemies and what to build to counter them is always important.
Kog'Maw's Caustic Spittle is his most essential ability for the first half of the game. The AS passive is important to keep up to be on par with any enemy champion who is building items for AS. I would max this ability first over any of the others.
Bio-Arcane Barrage is important later in the game for it's HP bonus damage, but at the start of the game it is used for harass. I would take W at level 2 and max it out second of Kog'Maws three basic abilities.
Void Ooze is not as reliable as his other skills when it comes to AD Kog so I take it at level 4. It helps to get away when getting low on health, or to chase enemy champions once the battle is in turn of your favor. I would max this last for it is not nearly as useful as his other skills.
Living Artillery should be maxed out when possible, just like any other champions ult. This ability should be used on the 40 mana use at all times to harass at bottom lane. It does great damage for cost of mana and should be practiced on champions to understand movement patterns. I would recommend saving mana by only using the 40 cost of it until you get a good advantage over an enemy, then using it between auto attacks once the enemy is on the run.
First skill is Heal to help survive any kind of push that the enemy team has the advantage. It should only be used as a self ability in the start to allow you to continue creeping. I know that sounds bad, but if the support can't get away, theres no need to try to save them and waste the ability.
Flash is a great second. I use Flash as a more offensive ability for I enjoy harassment in my lane. But Flash is a great defensive ability to use over ledges, cliffs, tree lines, anything that gets kog away from the battle. I also use it defensively when in team fights. As the ADC the enemy team will be (should be) going for you first, so flashing to the back of the team to lure greedy enemies or unaware enemies to your entire team.
Kog'Maw is my favorite ADC and, if used right, can crush any other ADC. Using kog with this build requires a lot of patience for the first 10-20 minutes of the game. Kiting, harassing, and creeping are a large part of this build to keep your money ahead of other champions. This build is somewhat difficult to do, because the BFS is expensive so it gives a disadvantage while the enemy ADC is building other items. Once the BFS is purchased you will enjoy your advantage over the enemy champion.
Hope this build is helpful and reliable!