Greetings, fellow Vayne player.
This isn't an experimental build, this is the Vayne build. This build will get you the optimal amount of usability out of Vayne, and will explain in detail why to take each item and in the given order aside of just saying "It works."

This is my first build, so sincerest apologies on the sketchiness, but I'll try to make sure to point out mistakes other Vaynes make, and how to play your role the best.

There isn't a whole lot to really put, in all honesty.

Looking at the pros, it's obvious that she's an incredible (if not the best) AD carry. She has one of the highest damage outputs, closely matched to Ezreal and Corki. She has incredible mobility with usage of tumble, and a pin that takes a bit of practice to use, but that's giving a disable to a carry... which is fair. Vayne's also the best chaser. She passively gets movement speed for following an enemy champion, so unless it's a Master Yi with Highlander active or a Miss Fortune with strut, you should be able to chase them with ease.

Browsing the cons, it's pretty stereotypical of an AD Carry, but if you have a brain they can all be compensated for. You are a CARRY. You are building for damage output, and not resistances. You will usually die if piled, so don't put yourself in that situation ahead of your team, you are to stand back and put out damage. If you are caught in a stun, you may die, or take just a lot of damage. That just means you have to pick up a Banshee Veil, because that will drastically improve your survivability with one item. And you're not invincible, so get over it.

Being an item dependent Attack Damage Carry champion, the better your farm and cs, the better your game will be. Of course even bigger than the amount of money you have, the items you invest in are even more crucial, but that will be saved for a later chapter.

Everybody knows that you must perform the killing blow to receive the gold for a creep kill. Sometimes, however, getting that last hit isn't always manageable. Sometimes two creeps can have their health dropping simultaneously, making you have to decide to take one or the other. However, Vayne gets the blessing of a cheap on mana ability that can be used to reset her auto-attack timer and get both of them. Assuming you took tumble at level one, you may be using it for harassment, but if not you can auto-attack one minion to get the CS then tumble into an auto-attack upon the other creep. It's really easy to pick up on. Make sure to get use of it when you have to clean up creep waves off turret, because every gold and creep kill gets you that much closer to another item, and rocking up your damage output.

"I can imagine the Flash, but why the Exhaust?"
I assume this is what you are asking yourself, because not everybody fully understands the purpose of Exhaust.

I'll begin pointing out the obvious, starting with Flash. You are a squishy champion, a VERY squishy champion. You will be forced into pinched off locations with no exit without intention, and sometimes another champion dedicates to killing you when they have the advantage. Flash is a lifesaver. Even beyond being a lifesaver, it can grant potential kills if another champion like Ezreal where to blink over the wall, or use flash.

Now for Exhaust... There are other viable summoner spells such as Ignite, Cleanse, Teleport, etc. but they will not give you the potential that Exhaust grants. A lot of people just use Exhaust when they notice it's off cooldown, assuming it's going to grant you a kill. Exhaust should be used sparingly, saved for a make or break scenario. Using it at a time when you know the kill is already guaranteed and they are helpless, that's a wasted summoner. Instead, you could use it when somebody tries to turret dive you, dedicates to killing you, or has an escape opportunity that Exhaust could prevent.

Ignite is not a smart idea. It has potential, it gives you that bit more burst on a kill, or the cleanup when somebody escapes with a bar of health, but it'll really not give you the same benefit you can get from Exhaust. All it takes is 1-2 auto-attacks to compensate for the loss of the ignite, which can easily occur in the duration of that Exhaust, on top of slowing them and nullifying a lot of damage.

Sprint is another viable, yet not recommended summoner. Your passive will let you give chase and the ability to dedicate, sprint will barely help you. It may make getting to a fight quicker, and that would still be kind of wasteful.

For the love of god, people need to understand that maxing Silver Bullets is NOT the way to go. You are gimping your damage output and wasting so much pressure you could be applying. One point in level 2 for the ability, that's all it takes. Maxing it out first is taking out points you can throw into Condemn for a ton of bonus damage and all the benefits it grants. That stun and pin is huge. It will devastate somebody when you catch them with it, and almost always grant a kill if not leave them crippled to the point that they have to recall or forfeit a death to you.

Second, you max Tumble. The reduced cooldown and bonus damage on it magnifies and makes it awesome for laning. You put one point in it at level 1 for the amazing harass it gives. When somebody comes up for an exchange of harassment, poke them with an auto-attack then lead into a tumble followed by an auto-attack as you retaliate back. Each additional point early game barely improves it, which is why you max it second.

Silver bullets? People just seem to max it for the pretty name. You put one point in it at level 2, so you can counter harass people hard. When they come in, you auto-attack, tumble in further, shoot and then once more to ignite the bullets and you pull off. You will win the exchange if you aren't sloppy, and they will learn to retaliate or they are going to die. But Silver bullets will never compare to the devastation of landing a Condemn pin.

Why Attack Damage instead of Armor Pen?
Your Attack Damage will make obtaining CS a lot easier, on top of making your combos more devastating and your auto-attacks (the bread and butter of early game damage) stronger. You can exchange the Quintessences for perhaps Armor Penetration or Health, but I guess that is up to you and you will be diminishing your capabilities. As for marks, there is no substitution that even compares. Seals don't give you too many options. You can exchange them for attack speeds if that's what you're into, but again, you will be dropping the potential you have with raw attack damage. Taking the mp5/lvl or hp5/lvl or anything else will not be noticed, that is like 100 hp at 18, and if you end up running out of mana on Vayne that means you are not going back ever or you are spamming Condemn for last hits. Glyphs have a few other substitutions, but I've come to realize that as you are a carry that extra magic resistance will be huge for you over attack speed or cooldown reduction.

Taking tank masteries is just silly. You are limiting your potential to take a safe route. If you are looking for a defensive AD Carry you might want to look into perhaps Teemo or Irelia, but not Vayne. You want to go into Offensive to get the extra attack damage. Going further gives you baby steps for each point you insert, where you can throw them into utility to get improved flash and neutral buff duration. It's not an experimental build, it's the build that will guarantee you the best potential with the Exhaust and Flash summoners.

Now, to get things straight.

Silver bullets isn't your damage, your tumble and Condemn is your damage. Stacking attack speed/on-hits will make you do a lot of damage, because you will have more Silver Bullets taking effect. Doing the work though, it takes 3 auto-attacks to set off Silver Bullets, and the maximum auto-attack speed is 2.5 per second. Now taking a look at Tumble, it resets your auto-attack timer. That means if you have a mind-blowing amount of bonus attack damage, you can land an auto-attack and follow it up with a tumble directly into another auto-attack. That is another devastating hit to take from a Vayne with a stacked Bloodthirster. If you don't take my word for it, then try attack speed/on-hits, then try attack damage and compare. If you are making correct work of your Tumble, I can guarantee you immediate results.

For the build, you want to lead off at level 1 with a Doran's Blade. A good early game will rocket you into an amazing endgame. So you want to follow it up with two more Doran's Blades paired with boots. If you know you are having a hard time exchanging harass because they have boots, then get it as your second item. If not, get another Doran's Blade or two before the boots. Doran Blades will give you a ton of bang for your buck, and will make you perform at your peak early game.

Following up the three doran blades, things become situational. If they are caster heavy, just a mess of Ability Power champions because they clearly all thought it is genius to do, then get a Hexdrinker. If not, you want to rush a Bloodthirster. Get the B.F. Sword first, then finish it. The Infinity's Edge will give you a slight more bit of damage over it, but the lifesteal makes it the better buy for that early. Follow it up with the Infinity Edge now.

At some point in time, you are going to need to upgrade your boots. Whether if it's right after you buy rank 1 boots, or after you finish Infinity Edge, you have to get them. When you do, if their team is a mess of disables and you know you're going to be eating them, then pick up Mercury Treads. If not, grab Berserker Greaves. Not Boots of Swiftness, you have a passive that gives you awesome chase, you want that bit of attack speed because it will ramp up your damage much more than Cooldown Reduction or anything else will.

The next stage of the build is going to revolve around getting Trinity Force. You want sheen first. Why Trinity force? Because of everything it gives, on top of it's passive. When you tumble and that auto-attack hits for roughly 1-1.2k, it makes them want to punch or beak something. In the buying process, always get the Sheen>Phage>Zeal.

Finally you will need to get Armor Pen. If you notice early that they are stacking armor heavy items, then pick up a Last Whisper before the Trinity Force, if not and they have light armor items or none, wait til now. If the carries have no armor items, and they tank perhaps has a Randuin's Omen, you want to pick up the Black Cleaver. This will make it so in 3 hits you are now doing true damage to them. If they are stacking armor items (ie. Madreds/Randuins/Thornmail/Sunfire) between the champions, then pick up a Last Whisper to get the most out of your other items.

Last, but not least, A defensive item. If they have literally NO disables, you can probably pick up another Bloodthirster, but very seldom will you ever see a team without one stun, fear, disorient, etc. So to deal with that so you don't just get stunned and die, pick up a Banshee's Veil. It will make life easier, increase your survivability a ton too. You could just be funny and grab another Attack Damage item, but unfortunately you can't deal damage when you're dead. :/