Hi and welcome to my Vayne guide.
Vayne is a ranged DPS assassin (marksman or ADC), she's big in mobility and surprise, pinning down single targets and severly weakening them before they can retreat.
Vayne is the best ad-carry in my opinion, she has invisibility with her Final Hour, huge mobility and chase potential with her Night Hunter which is tripled when you activate Final Hour and a stun (or a knockback) with her Condemn.
Tough Vayne is not that great early, being bullied by almost every other ADC, late she shows her true power, shreding tanks and 1v1'ing every other enemy, no matter how tanky he is, of course if she's played correctly.
In this guide i will teach you everything that needs to be known about Vayne, from the most simple things like the build to advanced tactics.

Pros

+ Hyper late-game carry + Great duelist + Great mobility + Tank shreder

Vayne is a hyper late-game carry, due to her Silver Bolts ability which makes her a tank shreder and a amazing burst-type ADC, she has great mobility, and she is a great duelist.

Cons

- Weak early-game - Mechanically difficult - Needs great positioning in teamfights - Range is kinda short

Just farm until you get your first items, you have no power early, you will have some chances if you have a support like Leona or Nami but most likely the lane will go in favor of the enemy adc early, but don't worry, just try to get into late game, then 'reck' them.

When you start the game you should always start with Doran's Blade and one Health Potion, i don't suggest you start with other items, Doran's Blade is the best offensive item early, giving you some damage, hp, and we can say it haves some lifesteal too.
First item you will need to rush is Blade of the Ruined King, because it's so good on 1v1's due to the active, which is sometimes crucial, it gives everything you need on Vayne : Damage, Att. Speed and Life-steal, and of course, combined with your Silver Bolts this item can shred tanks pretty easy.
Nex you will need to get a Phantom Dancer or a Statikk Shiv and a Last Whisper.This items will make you a monster, and from now on you will show your power.
Of course, every ADC needs this item Infinity Edge, it's very expensive but it's worth buying, it gives you 70AD, 25% crit chance, and it maximises your crit damage (from 200%AD too 250%AD.)
Make sure you grab your Berserker's Greaves when you think you need them .
For the last item i suggest you to go for another Phantom Dancer, but if you want to go with a deffensive item, you should go for Guardian Angel or a Banshee's Veil.

---

If the enemies do not have a tank, you can go for Bloodthirster instead of Blade of the Ruined King because it gives more on-hit damage(AD), but this will take away the Att. Speed and the awesome active.
Trinity Force is a pretty good item on Vayne, of course, because of the passives, you should get it when you have an easy game, or when you need more chase potential and more single target damage.
Sword of the Divine is an awesome item on ADC's like Twitch and Draven but it's not really worth taking it, you have better choices.
Black Cleaver is not for the ad-carries. Let's be serious, you have enough penetration with your Last Whisper, and the other stats do not really help an ADC.

Night Hunter[Passive]: It gives you 30 movement speed when you move towards enemy champions. Great passive for chasing an opponent, combined with Berserker's Greaves - Furor and Trinity Force, if you bought them of course, it makes you an unescapable monster.

---

Tumble[Q]: It gives you mobility and an basic attack reset. Great skill for farming, kiting, chasing ( specially if combined with Night Hunter and Berserker's Greaves - Furor), and of course killing. You can synergize it with your 3rd basic with Silver Bolts to get maximum single target damage.

---

Silver bolts[W]: It gives you bonus and true damage every 3rd hit on the same target. Great skill for shreding tanks and of course, bursting people down, synergises well with Blade of the Ruined King because of the % of the targets maximum HP.

---

Condemn[E]: It knocks the target back, and if they hit structures, they are stunned for 1.5 seconds and take bonus damage. This is Vayne's bread and butter skill, this is what makes her the best duelist, if you manage to stun the enemy, they can do nothing, so you have got some free damage on them, maximizing the burst potential. Can be used to escape a gank or a tricky situation too.

---

Final hour[R]: It gives you bonus AD, 1 second invisibility when using tumble, and it triples the passive's bonus movement speed. Exactly what i said, best duelist, this ultimate makes her unkillable and gives her huge mobility and chase/kill potential, duo to the tripled passive effect. You can win a 1v1 with 5% hp if you combine it with Condemn and use the invisibility wisely.

---

21 OFFENSE 9 into Defense OR Utility (#YOLO)

---

Standard masteries page for an ADC, if you do not want the deffensive stats you can put the 9 points in utility, tough it's not recommended. #YOLO

• Greater Quintessence of Attack Damage: You will need only 1, this will get you bonus damage which is great early.
  
• Greater Quintessence of Attack Speed: A must on Vayne, 9% more Att. Speed will help you at trades and farming.
  
• Greater Mark of Attack Damage: You are a ADC, you need damage, don't you?
  
• Greater Glyph of Magic Resist: You play versus an enemy support which most likely deals AP damage, don't you?
  
• Greater Seal of Armor: You play versus an enemy ADC, you need some armor, don't you?

As I said, Vayne's strongest phase is late-game, in late-game there are teamfights, so, how can you do good with Vayne in teamfights?
The most important things is to let your bruiser/tank to engage, because, if you will engage, you will die in 2 seconds.
Another important thing is to target THE CLOSEST target, not the ADC/APC, otherwise, if you go and try to kill the ADC, most likely, the enemy tank/support will protect him, you will get focused and killed. So yeah, just target the closest target, it's not a real trouble for you, you're a tank shreder, aren't you?
Activate Final Hour and shred the enemy closest target, kill them as fast as you can, then move to the next target, hopefully, your teammates will protect you and you won't die.

So, you got your core items, Blade of the Ruined King, Phantom Dancer and a Last Whisper, and let's say someone sees you pushing bot-lane and he wants to kill you, what do you do?
Pretty simple, activate Final Hour, auto attack as much as you can, and make sure you knock your target into a wall or a structure to stun him for 1.5 seconds, you will be able to burst him down fairly easy.
Can anybody really beat you in a 1v1? No, if you play everything correctly you beat every champion in this game 1v1, but you need to hit your stun and to use your invisibility correctly.
There are some dangerous enemies that can outplay you tough : Cassiopeia, Tristana, Twisted Fate, etc. But remember, if you play EVERYTHING perfectly, you will be able to kill every champ on this game.

So yeah, this was my guide on Vayne, hope i helped you somehow and you learned something from this guide and it wasn't a waste of time reading it.