Now this is just my opinion on how to play Vayne there probably are a lot better guides from a lot better players so feel free to leave suggestions and share you're own opinion. At the time that i'm writing this guide the most played, (and arguably the best,) AD Carries are probably Ezreal, Corki, and Graves, but Vayne is still a very strong pick. Vayne is a carry that scales very well late game and can probably beat out the majority of other carries past the 20-25 minute mark, so that being said Vayne's early game isn't as as a champion such as graves or corki. (unless you have a good agressive support.)

For Runes i like to build Armor Pen Quints, Attack Damage Marks, and Defensive Blues/Yellows. Another rune build people go are ArmorPen/AD for Marks/Quints with Att Speed Blues/yellows.

Greater Mark of Strength x9
Greater seal of Resilience x9
Greater Glyph of Shielding x9
Greater Quintessence of desolation greater quint of desolation x3

OR

Greater Mark of strength x9
Greater seal of alacrity x9
Greater Glyph of alacrity x9
Greater Quint of strength x3

Masteries are simple 21/9/0 I go 9 in defense just for that extra health and armor early on.

For items i like to get an early black cleaver just because the passive on B.C does very well with Vaynes W. Alternatively you can get your I.E. first but i usually do better with a black cleaver early on.
Core Item's
Bererker's Greaves- Attack speed

Black Cleaver- Good armor shred + Attack Damage + Attack Speed Good overall item

Infinity Edge- Ton's of Damage this is a must if you wanna do as much damage as possible

Phantom Dancer- Great attack speed, move speed, and the crit chance works very well with Infinity Edge

Blood Thirster- Fully stacked gives great AD and the life steal gives you a lot more survivability in fights

Last item is more situational

Banshee's Veil- Magic Resist HP and a shield

Quick Silver Sash- active is a cleanse that can save your life and Magic resist

Guardian Angel- good defensive item on most champions

Optional items for more damage

Madred's Bloodrazor- if the enemies have a lot of health

Last Whisper- good item and if you want i guess you can sell your Black Cleaver for Last Whisper but i wouldn't recommend it

Other Thoughts
One thing i see a lot of beginner players do on Vayne is rushing a phantom dancer, i wouldn't do that my self mainly because you have no damage and if i need attack speed i would buy black cleaver.
your last item i recommend to be defensive like a Guardians Angel or a Banshees but if you want to build more damage some good items are Last Whisper or Madred's Blood Razor.

One of the best thing about Vayne is the diversity of her skill build. Me myself i like to max out my Condemn (E) first just for that burst damage when you hit them against a wall, but i've seen many good players also max W first and some people even max Q for the cooldown. But for the most burst damage bot lane i recommend getting E cause its a safe skill that can save your life or get a kill. I max Q after E just for that cooldown and stealth (with the ulti) but you are free to get W for that extra damage.

Tumble- i max second just because in the laning phase it's not that great but in mid game team fights Tumble and constant stealth will be very rewarding.

Silver Bolts- I max this last (unless the enemy team is very tanky) true damage is awesome but i prefer Condemn and Tumble

Condemn- I max first it's stun and has great burst if you can use the wall to your advantage.

Final Hour- Makes you ninja nuff said

Flash- Do you have to ask why? Because it is amazing

Ignite- Early Kills :D

Cleanse- good choice if you know how to use it i always see bad Vaynes panic spam cleanse for no reason.

Heal- Eh you will get life steal later but if you want to heal bait noobs then go for it

Exhaust- if you want O.o but i would rather have my support get it

Vaynes team fight potential is very strong but her harass is not the best. Like i said in the introduction Vaynes early game isn't very good because she can easily be outbursted by other AD carries, but vayne with a heavy crowd control support can be deadly. Vaynes condemn paired with an Alistar Pulverize or Blitzcrank Grab can lead to an easy first blood. Vaynes shorter than average attack range means that you can be easily poked down by a caitlyn or trist so try your best to avoid their harrass and farm up as much as possible unless you can turn a trade into a full engage with your support.