I'm not the best player, but I'm someone with time and someone who likes Vayne enough to spend a while thinking about her. This is my first guide, so please take that into consideration.

Anyways, I hope you enjoy and find something useful!

(Picture by Mauri5io @deviantart)
Pros

Great chase
True damage for % health with Silver Bolts
Tumble can dodge some skillshots
Invisibility (Can be super useful)
A stun! Condemn (how many ad carries have that?)
can kite with Tumble.


Cons

Poor poke if used incorrectly
Oracles make you cry
Stuns make you cry
Running away isn't as fun as chasing
Like all ad carries, needs a support who knows how to play

Vayne is my champion of choice when fighting bruiser teams, or just teams with champions you know will stack armor (Like Malphite or Rammus)
I always want to pick Vayne when going against a Taric or Soraka bot lane, because I know the will be adding lots of armor to the team which gets pretty annoying as someone like Ashe.

The biggest thing is to know when NOT to pick Vayne

Vayne Does not do well against Caitlyn, Graves or Corki. In these cases, I would try not to pick Vayne simply because your lane is going to suck. However, if your team needs the true damage, then it is worth the bad lane. And you can still outplay them, it'll just be more difficult.

The best supports with Vayne

Pretty much everyone would agree Alistar works great with Vayne, but I'll tell you why. Alistar can line up your Condemn. Yup, that little skill that makes the match made in heaven. Alistar can use his knock up and headbutt to line them up for yet ANOTHER disablbe. This whole time you can be punishing them with your Tumble and Silver Bolts


Almost the same as Alistar, just without the heal. His grab lines them up, his knock up is just amazing, his silence prevents escape. But the reason we love him.. Because he lines up Condemn.


Nunu is different. This time the emphasis is on your Silver Bolts. His amazing attack speed buff works so well with it I'm surprised not so many people do it. Also his Blood Boil increases your movement, and who doesn't love MORE chase power as Vayne?


Taric is good. He stuns so you can Condemn. He adds a bit of armor. He heals. Just not as amazing as the other options, though.


Leona lines up your Condemn. But really, do we need more cc when Leona is in in the picture? Leona isn't the support I'd prefer, but still better than some of the other options.

Most of the time when I play Vayne I take Flash and Ignite.

HOWEVER!!!!!!!!
There are times when Heal or Cleanse would be a far better choice.

When to take Heal

First, look at your team. If your support can't heal, like Leona or Blitzcrank, or you wind up with someone random supporting you like a Veigar, then look at their summoner spells. If they don't take Heal, then you should think about taking it. It can be a lifesaver. You can bait a fight and make it fall in your favor. However, only grab heal if your support does not grab it. The support having heal in MOST cases is better, because then you can take Ignite to prevent the other ad carry from healing and give you the kill.

When to take Cleanse

This one you look at the other team first. If they have a champion like Morgana, who has a snair that would mean your death if you got caught in it, then DEFIANTLY take this spell. Also look at who you will be fighting in lane. If you're fighting a Taric, then this spell would be great even in lane to counter his stun. However if they have an Alistar, Cleanse Won't be as useful. It does not prevent knock ups or knock backs.

Why to take Ignite

5 extra AD isn't enough of a reason? How about reduced heals? No? What about free true damage while they are running away so you don't have to tower dive? Ignite Should be taken whenever you can.

ALWAYS KEEP FLASH

You can chase, run away, reposition and dive with this. No reason you shouldn't take it. NEVER substitute flash for another spell.

Now I know someone will say "Why no Exhaust??
Not good for you to have. Why? If you want this in your lane, have your support bring it and you bring heal instead. You shouldn't sacrifice your spell for something that could be JUST AS EFFECTIVE if your support brings it.

In most games I max Tumble First.
Why?
Because Tumble is great for chasing, positioning and kiting. Which in most lanes is what you're looking for.

But there are times when Silver Bolts is better.
When is it better?

Max Silver Bolts first if you're

1) Able to get off 3 basic attacks at a time. You usually aren't able to do this, though, because Vayne's burst isn't as powerful as some others (Like Graves) and you'll quickly find out you're loosing the exchange.
2) Your lane opponent has a lot of armor. For example, Taric (on the other team) Is taking most of the punishment and you want to kill him. True damage > Armor.
3) You need more burst. The way to burst with Vayne is simple but costs a lot of mana and isn't viable for a kill. You basic attack > Tumble into another basic attack> Condemn. This procs your third basic attack and you get the true damage.

Get a point in Condemn at least by level 3.

Why?
When used right, and in conjunction with your support, this is the way you get kills. This is the way you escape (You push them away). This is one of the simplest skills that will win a fight for you. If you've played Poppy, then you already know how walls are weapons. If not, then take it from me. Walls are Weapons.

Use them wisely.

Final Hour

Strange how I haven't talked about her ult yet, isn't it?
Well it needs its own section.

Final Hour Adds attack damage, stealth when using Tumble and also buffs your passive, giving you even more chasing power.

This ability is the reason Vayne is an "Assassin". And it's probably the reason I love her. Vayne can use Final Hour to line up her Condemn and the enemy champion won't even realize it is happening. Then once they do, half their health is missing and they've got to run. No biggie, your movespeed buff is so legit that you can catch anyone (Except, say, a Nidalee).

Now here is the big question.

When to use Vayne's Ult

Use her ult only if you are sure you're going for a kill. It is called " Final Hour", after all. Do not engage a team fight with it right away, because you may end up wasting it and the bonus AD, stealth, and movement is something you DO NOT want to waste. Make sure the fight is really going to happen and turn it on.
Its use in 1v1s almost always makes Vayne win. The stealth from Tumble can confuse your enemy into missing skill shots, or running the wrong way. They'll be Boxing with Shadows. Make sure to switch directions with your tumble. Don't be predictable.

Ult when you see someone alone. Hide in the bushes and then ult, Tumble out, and Condemn them into a wall, or at least away from where they would be running away. The extra ad, and the element of surprise, can get you a kill even on someone who beat you in lane.

When in a Team fight, you use your ult to prevent yourself from being focused. If someone is on top of you, you go invisible and take a step back and keep attacking whoever you can. This isn't a get-out-of-jail free card, though. Tumble only gives enough stealth time to kite, not to escape. It doesn't work like Wukong's or Talon's stealth. Though if you Tumble in the way they don't expect, it just might be the escape you're looking for.

A well played Vayne can burst. It is possible.
If you want to play burst Vayne Then learn her combo
Basic attack > Tumble into another basic attack> Then Condemn (preferably into a wall), but the real reason to use Condemn is to trigger Silver Bolts.
If you DO hit them into a wall, then get off at least one extra basic attack. You might even be able to proc Silver Bolts a second time.

This works extra well if you're going the path of the Trinity Force.
If you do build Trinity Force, then you should sub out Phantom Dancer for it.

Vayne can be built many ways and still be effective simply because of her Silver Bolts True damage. So keep this in mind, Attack speed is useful. Screw damage if you have to. Attack speed will proc your true damage, keeping your dps up even if you have to build more tankie in a game.

The most amazing item to have if you didn't take cleanse:

~QUICKSILVER SASH QUICKSILVER SASH~

As Vayne get Quicksilver Sash If you didn't take Cleanse. Not always necessary if you aren't being focused, but if you are, get it. Especially if your opponents have someone who likes giving you a hard time and leaps onto you at the start of every fight like Jax. If they have a Warwick, a Malzahar or a champion like Morgana Then get this. Don't argue, get this. Think about getting it EVEN IF you have Cleanse and are fighting them. You don't want to be the reason your team loses, and their snares can make you die. Or at least make you useless for a while. And any amount of time that you're useless is far too long.

Problems with a caster? Look under here

Some casters can kill you in one burst, making you unable to assist your team AT ALL during a fight. Sometimes you die before you get in a single attack. Casters like Veigar or LeBlanc do this to carries all the time.

Solutions!
As I said before, Attack speed is more useful than attack damage. The top two items I would consider if a caster is giving you problems is a Wit's End and a Maw of Malmortius. These two items will keep you doing tons of damage and give you the survivability you need.
I would sub Wit's End in for your Phantom Dancer and Maw of Malmortius in for Black Cleaver

Also consider a Frozen Mallet. This will help you survive bursts while giving you even MORE amazing chase and a bit of damage.

Next consider getting Mercury Treads. This small switch can do great things for you, especially if you don't want to get a Quicksilver Sash.

Bloodthirster. Wait? What the hell? Hush, just let me explain. If you have lifesteal, you can come back from almost anything. Get some MR, too, though. Don't come crying if you die with 3 bloodthirsters. That's not what I'm saying. I'm saying that IF You're burst, and you're not dead, then Fear not! Because ap carries have just that, a burst. Nothing more. If you survive their burst then you can heal back up with your life steal while they are on cool-down and continue helping your team.

Problems with a Tank? Then look here

First off, Why are you being focused by the tank? Your positioning was horrible. However, even with proper positioning sometimes you can be jumped by a Jax or a Nocturne and put in a bad place.

Solutions!
Again, Quicksilver Sash will be a lifesaver. Most tanks have cc (Duh) and that is what kills you. If you can counter Jax's stun, then you can continue attacking him and continue healing yourself with your lifesteal. (Again, Bloodthirster IS a survivability item, but not a burst-prevention item.

Next, tanks have armor. You have to shred this in order to kill them. Think about buying an early Black Cleaver or Last Whisper. Buy the Black Cleaver 99 times out of 100, because you're Vayne. Vayne should attack the same target 3 times (the max Black Cleaver stack) to proc her Silver Bolts anyways, so it is almost always better. Buy Last whisper if you're not able to get off several attacks for whatever reason.

Maybe they build too much armor, not enough MR and your caster doesn't do enough. Then think about getting Madred's Bloodrazor. This will add even more total health damage, but unfortunately, it is reduced by magic resist. If they are also stacking Magic resist, then this item isn't as useful as it claims to be.

Last item I'd suggest is a Trinity Force or a Frozen Mallet. They both give a bit of health and a SLOW. The slow will let you kite for days. If he is still fast enough to keep up with a 2 second Tumble, a Condemn and a slow, then your team isn't doing its job of protecting you.

Problems in lane?

One of the hardest things in a lane is sustain. If your support can't heal, and theirs can, you'll find yourself backing. Which will make you under farmed, under leveled and, sooner or later, dead.

Solutions!
Early life steal is great. Get Doran's Blade and Vampiric Scepter as soon as you can. Even if you have to back early to get it, it might be worth it. When you have low hp you'll realize you're being zoned. You might be better off backing.

Early armor? What the heck? I'm an AD CARRY. Shut up, You'll build a guardian angel out of it late game. Just grab a Chain Vest and call it good.

Lastly, this one is the weird one, build a Sheen. If you can win the harass exchange then you'll gain control of your lane. The Sheen can be sold later, or built into a Trinity Force (Which is controversial, but I LOVE IT.)

Lastly, if you're still losing, then LOSE YOUR TOWER. Let your tower die and then you'll deny them farm and kills. The 150 gold (For everyone..) from the tower might be worth it, because now their AD carry can't farm, just like you.

Almost forgot to go into this. This is what makes a good ad carry good, and a bad one bad.
You have to know when to attack, and when to back off.

There are a few keys to knowing where to be.
1) If you're last hitting during a stand off, then NEVER go beyond your tank or some other melee. Miss the farm, it's not worth you dieing.

2) Do not poke. That isn't what Vayne does. If you're trying to poke, then you're getting in range of their abilities. Malphite could jump you, so could Irelia, Wukong, Fizz or pretty much WHOEVER on their team feels like it. Let your team poke.

3) Wait. Make sure the initiation has been used. If they have an Amumu, then take a step back and make sure you aren't the one he gets his Bandage Toss on. If you're caught in his ult, then that's why you should have brought Cleanse or purchased a Quicksilver Sash. Anyways, wait this first part out. RIGHT after the other team chooses their targets, you roll out the big numbers and start killing.

Tips

1) Never extend past your team to attack the other team's fed ad carry or ap carry. Your whole team can yell "We need to focus Caitlyn", but that's not your job. You need to do damage to whoever you can while staying safe. If their Rammus is the only person within range, then you ARE going to attack Rammus. If you get a choice, like their ad carry is extending too far, then DEFINITELY go for them. But you do not want to get caught in the middle of their team just because yours told you to focus their carry. No. You must live.

2) Let your team die. Yes, be the person who leaves someone behind if they were 5-man ganked. You're more important. If it is just one, or two, then maybe you can help. But never risk yourself for someone else. You're the most important team member to be alive and in a fight. Never let yourself die for someone else.

Vayne is great if you're looking for true damage, mobility and a fair ammount of cc. If you're looking for a ranged ad carry that can make plays, then Vayne is ready for you to try her out.
Enjoy the Shadow Boxer