Early game focuses on 2 things - CSing and trading. As Vayne, you're gonna want to focus more on CS, due to your short AA range.

When killing creeps in the early game, don't try to kill with 3 stacks of W. W does not do much yet to the creeps and takes too much time to focus on one minion. Farm like a normal ADC would, making sure to last hit. After level 2, I secure the Cannon Minions with the occasional Q. Its just a little thing I do, and by no means necessary.

You're Q and E and going to be your harassment tools. If you're playing against someone with a higher AA range, like Caitlyn, you can surprise them by using Q to get close, auto attacking with the little extra damage from Q, then Condemning them (E). You can win a large amount of trades this way since you can catch them off guard, especially if they're last hitting a minion.

During the early game however, you're Q is going to be most useful for dodging enemy skill shots, and should be used primarily for that purpose.

A handy tip I use for Bot lane Blue side, when pushed to turret, is to pull the minion wave up slightly towards the tri bush. You will take a slight amount of minion damage, but it should be negligible due to the Dorans blade start. The turret won't be in range of the minions and the wall should largely protect you, or give you time to escape from, enemy skill shots and auto attacks, due to the distance it provides. Otherwise, if an enemy comes to follow you, you can easily Condemn them (E) into a wall and stun them

Blade of the Ruined King is where you start to snowball and get aggressive. This is where Vayne starts to become a god. The new combo is Ult > Q > E > AA > BOTRK > AA and all following auto attacks. This will be your combo from here out. Blade of the Ruined King just works so well with Vayne. Attack speed, lifesteal, the attack damage, and active just synergize with her kit.

Botrk is easily your most important item, and you should feel pretty confident after you get it. You should then get Statikk Shiv to help with dueling, and because of your high attack speed, you should activate its passive often.

Your 2 mid game items are Zephyr and Phantom Dancer. After the completion of Phantom Dancer, you are pretty much in late game, as the other 2 damage items are just extra.

You may have been wondering, "when do I buy boots? I didn't see it in the build". Well, you don't. The passives from Statikk Shiv, Zephyr, Phantom Dancer, and your passive more than makes up for boots.

Now you are going to try to catch people out of position, near walls. Just do the Q>Condemn combo or Flash>Condemn combo and vaporize them with Botrk and your ult. One of my favorite memories was when I flashed over a wall to Condemn the enemy Varus and destroy him. He didn't even get an AA off.

Only thing you have to do now is keep yourself alive, because you can kill practically anyone 1v1. Outplaying is one of the most important parts of playing Vayne and that relies on Q, E, and ulted Q. With your attack speed, its more worth it to not Q in a straight fight than to Q, so use it to chase and avoid skill shots, as well as make getaways. Flash can be used to outplay your opponent as well, you can catch them off guard by Qing at them, flashing to their opposite side and comboing them.

Make sure to kite people as well, and get off your W, the most important part is your W. Its why you (usually) don't need a last whisper, and does tons of true damage with your attack speed.