Why would you get vamp scepter first? Any corki that is autoattacking creeps is not playing to Corki's strengths. Corki is all about last hitting, harassing, and chasing. This means you are NOT autoattacking. You attack each minion once. you want the other champ to be pushing up on your turret so you can kill him with a q + e + w combo as you chase him back to his tower. Get a dorans shield and harass with autoattacks or get a mana crystal and harass with Q. When you hit 6 harass and last hit at the same with his ult. Dont use Q to do this much, its not mana efficient.
Why would you buy a bloodthirster? If you are any good, you are gonna get focused first in any team fight, or ganked first when you are laning. When you die, you lose all the stacks you had. Black cleaver is the better option, it synergizes with his E. If you are getting hit during a teamfight, it most likely means you got too close and are getting gangbanged by the entire team. No amount of lifesteal is gonna save you in this situation.
Since you recommend valking in, now you have no escape if you get CCed. Your BT stacks are gone now. Don't valk in, always save it for escaping. Get flash, it will help you escape those seemingly impossible situations. Exhaust will not help you run from 5 pissed off champs.
Build Merc treads or Boots of Swiftness. Escaping alive as corki takes precedence over killing a champion. Every time you slip through their fingers, the more irrational they will play, in their greed to finally kill you.
Build any part of triforce first, it will help you more early game than a mid-late game BF Sword. I am going to assume you are bad at last hitting since you buy a vamp scepter. You should have enough gold when you hit level 6 to buy a sheen if you bought a mana crystal. Zeal and phage are also viable options.
Your guide is focused more on killing individual champs. Corki excels at destroying whole teams at once, thanks to E + Black cleaver casters die faster than they can faceroll their keyboard, tanks melt with no armor, any physical dps is ruined by Q. Champions running away? Kill them with R.
I dont know how many times you die a game, but I average about 3-4 with about 15 kills/assists per game as corki. This is not to say how good i am, but how good corki is as a champion period. Flash + W + Boots allows escape from almost every situation, Q, E, R are all amazing AOE attacks that are good early game and get better late game.
He's an in and out type of champion, with good burst and good caster abilities.
This turned into a really long post, but corki is so much more versatile than just an autoattacking dps. He is one of the best champions in the game imo, and he doesn't even have a CC.
If you didn't read any of this and skipped to the end here's an item build for you
Sheen (get liz, kill someone)
>Rageblade, Last Whisper, Rylai's or any health item if you are getting focused a lot
> A DPS item to finish pushing their towers/nexus. Phantom Dancer is a nice one. Dodge helps a lot.