Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
This is primarily a crit build, but it has a strong emphasis on attack speed and damage as well.
Personal preference matters a lot here but you definitely want something that is going to increase your survivability in the early game. Heal and Ghost are both strong choices, but in the late game, Heal's utility begins to fade while Ghost is still strong since it will help you escape ganks and chase. Conversely, Heal keeps you in your lane longer in the early game, which will help you level and build faster. You want Teleport as well, since it will help minimize time away from your lane. It's also a great offensive tool in the late game.
You want to go up through offense until you get the 10% bonus to crit damage. The remaining points are really personal preference but you have just enough left to get the XP bonus in utility so that is what I usually do. Every little bit helps, especially since Kog really doesn't start to shine until mid-late game.
Pure crit damage here, unless you want to mess with the item build. In that case, crit chance runes works as well. I think you will get a better return with all damage here though.
Start with Caustic Spittle and alternate with Bio-Arcane Barrage until your ult is available. Keep alternating until you've maxed them both and finally finish with Void Ooze. Void Ooze is only really useful as an escape mechanic since you don't build up AP. Living Artillery isn't going to be doing a huge amount of damage in this build but it's great for scouting and for finishing fleeing champs.
At the start, you have 3 viable choices: Long Sword, Dagger and Brawler's Gloves. My personal choice is the Gloves. They can build into a Zeal or an Avarice Blade, which gives you some flexibility. It would probably be best to build up a Dancer first but if you are struggling with your farming, the Avarice Blade may be a better pick. After the first Dancer, you definitely want Infinity Edge next for the damage bonus and crit damage increase. Follow with another Dance and that will put you at 80% crit chance. You want 1 more crit chance bonus item- 2 if you want to make sure that absolutely everything is a crit. My personal preference is Youmuu's Ghostblade. It is one of the few items that offer a damage bonus along with a crit chance increase. Its active provides both an escape and a chase mechanic and the armor penetration is a nice thing to have. Depending on how your game plays out, you may want to pick up the Ghostblade at the Infinity Edge position since it's cheaper. Atma's Impaler and Trinity Force are not really recommended, since you won't have enough health to make Atma's damage bonus valuable and Trinity Force provides you with some redundant/useless bonuses for this build. Frozen Mallet should be included since it provides you with some much needed health and a slow in addition to the +20 damage. At this point, it really depends on how the game is progressing. You can't go wrong with anything that buffs your physical damage. Black Cleaver is a great choice as is a Bloodthirster. Tiamat makes you a farming beast. Manamune is an interesting possibility, but only if you rush it since you need to max the bonus mana to make it viable. At level 18, assuming you hit the bonus mana cap, you will be doing 20 + 46 (from mana) extra damage. Most games will probably not run long enough for this to be effective but it's an interesting option if you really want to spam Living Artillery.
It's worth pointing out that if you get up to the Frozen Mallet, you will be doing roughly 1420 damage per second. The other team may counter-build by buffing armor so at some point you may want to look at Last Whisper for it's armor bypass. Also, you might want to skip buffing your attack damage with the last item and instead go for some MR since most champs will try to stay out attack range and just hit you with abilities. Force of Nature is worth a look if that is the case, as are Mercury Treads and Banshee's Veil.
A Note on Madred's Bloodrazor
Yes, this build does not have a Bloodrazor. While it is synergistic with Bio-Arcane Barrage, your target will need to have a stupid amount of HP to make it as effective as a pure damage item in this build. For example, a Black Cleaver does 75 base damage. With crit (305%; the calculation above doesn't seem to include the +10% from masteries) it does 228 damage. The Bloodrazor will do 91 (30 * 305%) and then the magic damage bonus, which isn't included with the crit calc. The on-hit will have to do at least 137 damage to match, meaning the enemy champ will need to have at least 3425 HP. This is by no means an unheard of amount of health but it is on the high end of things for non-tanks. If you are targeting a tank with a huge amount of health, odds are pretty good that he has buffed his MR as well, making the extra magic damage trivial. Plus, there is the matter of which item to swap for the Bloodrazor. You can only really replace the Frozen Mallet and/or your last item without breaking the build. I think the Mallet has a much greater upside with its slow and health. I can see a Bloodrazor as a possible last item in a few (very few) situations, but for the most part a Bloodthirster or Black Cleaver are far superior choices.