(This is my first guide ever. I hope it works for you. It works for me. Also, sorry mine's not pretty and doesn't have pictures. Just deal with it. It has the information.) (:
This is a chapter on why I like Vayne. Not only does Vayne look cool and have cool lore, she's fun to play. Vayne is like the Batman of League of Legends. Both of her parents are dead, so she's fighting evil to avenge them. She's also extremely awesome in combat, if built correctly. If you're playing Vayne and become fed, there's nothing the other team can do to stop you. You are almost invincible. And here's why; Vayne's W ability, silver bolts. Vayne's silver bolts EASILY make her one of the best ranged carries.

I honestly don't want to get extremely technical with these. Not that they aren't important, but I would just like to keep this part short. For marks? Armor penetration. For seals? Armor. Glyphs? Maybe a nice combo of cooldown reduction and magic resist (or one or the other.) Quintessences? Life steal is good and/or movement speed.
Honestly, I think you can use your own good judgement for masteries. Attack damage, armor pen, good defensive runes, etc. Just typical ADC masteries will be fine.

The correct items are ESSENTIAL for Vayne.

Starting Items

The starting items for Vayne extend to three possible items. Boots&pots, Doran's blade, or a long sword&pots. Go for the Doran's blade. Although it's argued that pots help maintain a lane longer, the Doran's blade restores 5 points of health upon every basic attack. Which may seem small, but actually scales well with Vayne's small amount of health to begin with. The extra 10 attack damage helps keep the enemies distant. Boots are absolutely unnecessary right now. You are not jungling, and you are not yet roaming. Starting with boots seems foolish to me.

What to build next

Many guides for Vayne that I have seen offer heavy attack damage. This is foolish.
The charm to Vayne is the true damage she deals every third attack. Next on your shopping list? A recurve bow. This recurve bow will help utilize Vayne's heavy early game damage and third attack true damage.
Once you have the money, build your recurve bow into a Runaan's Hurricane. The additional attack speed this grants you is whopping, and allows you to attack multiple targets. Farming for Vayne couldn't get easier. This will help you gain some money.
Now that you have the money and attack speed, buy Berserker's Greaves. This will give you a bit more attack speed, and essential movement speed for wandering or sudden team fights.
Now money should be easily accessed. Buy a Brutilizer. This will give you some extra damage and some nasty armor penetration. Build the Brutilizer into a Black Cleaver for extra damage and (much needed) health.
At this point, you should seem pretty scary. Sell the Doran's Blade; you don't need it anymore (if you haven't already sold it.) Build up a Statikk Shiv. It doesn't matter if you build a Zeal or an Avarice Blade first, whichever you find fitting to your game. But the extra electric charge, along with the attack speed, aaalong with the true damage will shred your enemies (and will absolutely destroy minion waves. Going broke won't be a problem for you. In fact, ask your jungler if you can go swipe red buff and destroy in a team fight.)
You're not quite invincible yet, although you're close.
The final item you should get is debatable. If you're having trouble surviving, build a Bloodthirster. This will give you a crazy attack damage boost, and the life steal needed to survive for quite some time. However, you can go for a classic item: The Phantom Dancer. This, along with the Statikk Shiv, will make almost all of your attacks critical hits, and they'll be so rapid that it won't matter. That along with every third attack being true damage, and the extra bolts from Runaan's Hurricane? Yeah. You're pretty much invincible now. Go get a pentakill, plox.

This will be a short chapter.
Barrier: Eh, probably not.
Clairvoyance: Absolutely not. You are not support.
Clarity: Don't even think about it.
Cleanse: Eh, it's okay, but I wouldn't recommend it.
Exhaust: Yes, this is a good one, only if you're feeling very confident.
Flash: I think flash is essential. This and the tumble ability make Vayne impossible to catch.
Ghost: Meh...This once is okay, but I wouldn't recommend it.
Heal: Heal/Flash is the way I always go. It's good to have in a sticky situation.
Ignite: No. You do not need it. You can snipe the enemy or throw in a few last shots. You do not need ignite.
Revive: Revive is okay, but again, I wouldn't recommend it.
Smite: Don't even think about it.

Passive

Night Hunter- Vayne gains an additional 30 movement speed when chasing an enemy champion. This is nice when chasing a champion who is almost, but not quite, dead.

Tumble

Vayne tumbles a short distance. After the tumble, her next attack (depending on her level) does an additional 30-50% damage. Which is awesome for ensuring kills. Tumble is also awesome for escaping. Though tumble does not work through walls, it still give you more distance from the enemy.

Silver Bolts

Vayne's coolest ability. Every third consecutive attack deals true damage, based on percentages of their maximum health. Dealing 30-60 additional damage, depending on the level. For those of you who do not know, this is a nasty, nasty passive. Nasty for you enemies, at least. This is what will help you shred enemies, no matter how much health they have.

Condemn

2 This is a fun ability. Vayne takes the large crossbow from her back, firing one bolt at the enemy champion, dealing 45-185 (plus 50 AD) depending on level and knocking them back. If you use Condemn and the champion is knocked into a structure or wall, they are stunned for 1.5 seconds and an extra 45-185 (plus 50 AD) are dealt. A pretty scary ability, if you ask me. This ability is extremely hard to use. When I began to play Vayne, I never used it. Now, I use it as soon as an opportunity is opened. A common assumption is that enemies are only stunned when you knock them against walls. This is false. They will be stunned if you knock them against any structure (yes, including turrets. Your turrets, specifically.) This attack is also great for escaping. That pesky champion right on your tail? Shoot them straight back with a quick click of E. Condemn has proven most useful either early game, or late game in team fights. Those spontaneous team fights in the jungle are perfect for condemning. There are walls everywhere. Use this and your ulti and potentially sway a team fight.

The Final Hour

The ulti. This ulti is extremely easy to use, has a relatively short cooldown, and puts Vayne into stealth after she tumbles. This attack temporarily gives you 25-55 bonus attack damage for 8-12 seconds. Also, after using Tumble, she goes into stealth for 1 second (more of an advantage than you realize, yes, I know it doesn't seem like much)
The great thing about Vayne? Any of her attack in any order can be a kick-*** combo.

I put these two into one category, because they almost work together.
As Vayne, with your Runaan's Hurricane, farming is easy for you. However, be considerate, and try not to be greedy. Though, if no one else is taking that huge minion wave, go for it!
The same applies to the jungle. If you have a jungler, always ask before taking things. If you don't have one, or if they don't care, get red buff later game. This will burn the enemies you hit. It's nice to have as Vayne.

Laning

As Vayne? You're probably going bottom with a tank or a support. Always lane with a Volibear or a Blitzcrank if at all possible. Blitz's grab and Voli's throwback are awesome combos with Vayne's condemn. Vayne is also so squishy, that she needs someone to initiate and take some damage for her. Though a support works fine, I always prefer a tank. Both are sufficient, though. Just as long as they are AP focused.

Roaming/teamfights

This should be to the best of your judgement. DO NOT TRAVEL ALONE AS VAYNE. Unless you are going through your own jungle to a different lane, BE CAREFUL. You are SQUISHY. If you run into a few people hiding in the jungle, you are doomed. If you have an escort, you get a free double kill. Keep this in mind.
For team fights? Stay in the back and pick off their squishy carries. The priority is AD carries first, then AP carries, and then snag whatever is left (support and tanks.) Go for that ace (: