Corki is a great champion that can deal massive damage, and lower their armor for your dps allies to help finish them off. This is all based on my opinion of what corki is, and this guide also depends on runes, as well as how well you play a ranged burst damage champion. This is also a guide for laning without soloing, i don't like to solo with this champion because usually there are other champions that could use it more.
Runes
i always prefer per level, because with the items i get at lower levels lasts long enough to get make it more useful than flat):
Crit damage Marks (makes your auto-attacking with infinity edge and gatling gun active very lethal)
Health regen per level Seals (The regen allows you to stay in the lane longer, i will explain later why i get regen instead of health for survivability)
Cooldown reduction per level Glyphs (the cooldown reduction will allow you to cast gatling gun more often, and that way you dont have to waste inventory slots on cooldown reduction, since this and blue buff will suffice)
Crit damage Quints (the bonus crit damage along with the mastery and infinity edge gives you nearly 300% crit damage, and with reduced armor and armor piercing mastery even the most heaviest character's will be taking massive damage)
In some cases i thought of trading crit damage for armor pen, but sometimes it doesnt seem necessary, but like i said earlier this is all from my perspective of playing corki.
Masteries: i prefer a 21/6/3 build, focusing in the attack damage aspects of offense, the armor and MR in defense, and Health and Mana regen in utility. The armor piercing talent is my favorite, because with gatling gun most normal champions wont be reducing any of your damage.
Summoner spells: My favorite spells are Flash and exhaust.
Flash along with Valkyrie makes it almost impossible for you to not get away (except for stun locks of course) Also Flash can also be used to initiate and Valkyrie to escape.
Exhaust is great for making you more effective than their DPS champions, because even if you're exhausted gatling gun can still be reducing their armor and dealing massive damage to their entire team. Exhaust also can be used to keep them in your gatling gun, and for survivability.
Abilities: My all time favorite ability corki has is gatling gun. it can be used anywhere from team fights, to pushing, and to farming, and with the armor reduction it can turn the tide of battle. With that being said, it should be obvious i focus in gatling gun.
Ability order:
Gatling Gun
Valkyrie/Phosphorous Bomb
Gatling Gun
Phosphorous Bomb/Valkyrie
Gatling Gun
Missile Barrage
Gatling Gun
Valkyrie/Phosphorous Bomb
Gatling Gun
Valkyrie/Phosphorous Bomb
Missile Barrage
Valkyrie/Phosphorous Bomb
Phosphorous Bomb/Valkyrie
Valkyrie/Phosphorous Bomb
Phosphorous Bomb/Valkyrie
Missile Barrage
Phosphorous Bomb/Valkyrie
Phosphorous Bomb/Valkyrie
I prefer one level in Valkyrie and Phosphorous Bomb at the start because they are both very useful, Phosphorous Bomb can be used to search bushes and to see invis, as Valkyrie can be used to escape from any situation. I prefer Phosphorous Bomb at level 2, but if your having trouble in the lane then Valkyrie would be more useful.
At level 8, you should level up whichever one is more useful in your situation. If they have a character that can go invisible, then Phosphorous Bomb would probably be the most helpful, but if they have no invis capable champions then Valkyrie would most likely benefit you more. Like i said earlier though, its all on your situation, how well the game is going, and your opinion on his abilities.
Item Build: My preferred item build centers around damage more than anything else, he has enough base health you dont have to worry about it too much, but if the game is going terrible you might have to improvise and stop the main build and add in some health items, but with Flash and Valkyrie survivability shouldn't be the most important thing on your list
I always start off with an elixir of fortitude and a health/mana pot. The elixir gives you life to be able to stay in the lane and survive early game, and it also gives you enough damage to make your Gatling Gun stronger.
Now depending on how well im doing in the lane and when my next trip back to the fountain is, decides on what to get next. If im not doing well and i need some support because the elixir just isn't doing it for me, i grab a doran's blade and tier 1 boots, along with possibly another elixir.
If im doing well and can manage being in the lane without any immediate support, i buy my tier 1 boots, start saving, and head back to the lane. (you may also upgrade the boots, but the upgrade depends on the opponents team. if they are caster heavy with disables, merc treads is the way to go, in most other cases berserker's greaves is the right choice because the speed will assist you later)
After you have enough gold, buy your BF sword (i prefer buying BF sword first because its harder to get gold later game if you dont have that kind of damage, whereas its easier to farm with less items earlier on) for infinity edge (if you are still having trouble and won't be able to farm for it, you should buy a tiamat, it will give help you long enough so that you may get your BF sword)
If you havent upgraded your boots, now is the time, but for everyone else you need to finish up your infinity edge.
If your still doing well and think you can make it to another BF sword, go for Blood thirster (the lifesteal along with the massive bonus damage it can give makes you very useful in taking out their squishier champions), otherwise grab a Tiamat now.
After bloodthirster it should practically be game either for your team, or theirs. But if somehow the game is still going, and your able to take out the squishy champions but not the tanks, then Black Cleaver is the item for you, making your armor reduction even more godly.
After Black Cleaver is practically open game, you should grab whatever item you need to make you more useful for your team. Tiamat's splash can help destroy their squishy champions all in the same crit, but if you need more survivability than damage at this point than Banshee's Veil and/or Guardian Angel are also very useful. (guardian angel can allow you to let Valkyrie cooldown to give you time to escape on your second chance, so that you wont loose Bloodthirster stacks)
This is all from personal opinion, and based on how good your teammates are. Trust your lane partner and work with him to make your armor reduction as useful as possible. Tell me how this guide works for you, leave some feedback, and feel free to tell me your opinion. Feel free to throw me an invite and play a game or two with me.
Goodluck in your games, and have fun. I hope this guide helps and i hope to see you all someday in game.