Vayne is, in the right hands, a very strong ranged champion that serves as an assassin, a ranged carry, and is hard to catch if the player doesn't do extraodinarily idiotic things (Which sadly, I am prone to doing). You will be assaulted by a TLDR wall of text because I have no idea how to put images in [first guide].

I go 21/0/9 masteries because Vayne shouldn't even be able to be touched early to mid game if you have map awareness and don't over extend. In the case that you are focused- well, Armor and MR won't save you if the enemy team is coordinated enough to even designate a focus target. For runes, I tend to take 9 ArP marks, 9 mana regeneration glyphs, and 9 attack speed seals. The only attack speed for my build you'll be getting until late game is Trinity's, and Youmuu's Ghostblade active, which is very effective with Vayne if you remember to use it. For Quintessences, I either take ArP, Attack Speed, or Movement Speed (Which will make Vayne harder to kill, or easier to pursue enemy champions). I would recommend ArP, but it's mainly preference, just like this guide is meant to inform you how I play Vayne.

For items, I start Vayne off with a Doran's Blade or a Vampiric Scepter. VS will ultimately give you longer staying power and if I go that route, I tend to upgrade to Emblem of Valor after Berserker's Greaves. Stark's Fervor is after Infinity Edge.
Trinity Force is a very effective item on Vayne (doubles your tumble damage, gives you extra damage if you manage to hit someone with Condemn to finish them, and the other passive stats are swell as well) and I usually pick up Sheen if I have enough gold after returning to buy boots. I typically don't, so buying items along Sheen's line will not affect your stats (You don't use AP and you have mana regen runes) and in higher level matches, that may prove to be fatal. Longsword is what I usually buy, then upgrade to a Phage I will buy Sheen when I can, but unless I can afford the whole thing in one go, I skip it.

Youmuu's Ghostblade is effective on any ranged AD in my opinion early or late game, as it gives escaping power with its passive, extra attack speed before Stark's Fervor and after it. The damage and critical strike (also ArP) you will benefit from early - mid game, as you can pretty much upgrade each time you return to base instead of farming up for an Infinity Edge. The rest of the item build is pretty much self explanatory, Stark's gives ArP, AS, and Lifesteal, both to yourself and teammates. If a teammate already has Stark's, however, I would suggest buying a Phantom Dancer. The last item in the build is purely situational- If I feel I have enough damage (And believe me, you will hit like a truck even without another damage item), I will either go for Last Whisper if their carr[ies] are starting to build up or already have built up against me, or The Bloodthirster. Banshee's Veil or Thornmail
for heavy AD or MR teams. If the enemy team has heavy CC, I like to buy Cloak and Dagger because of tenacity (I prefer not to buy Mercury Treads because Vayne is stunted on ASpeed early game.

Flash and Exhaust are my preferred summoner spells. Flash can always get you out of a jiffy, but it's highly situational....for most champions. With Vayne, Final Hour and Tumble combined with Flash will allow you to make clean escapes. Relatively speaking. I say this because if the enemies have surrounded you or have perhaps, Jack in the Box set up to jack you as you run (no pun intended) you are dead seeing as you are a squishy.

Exhaust is good to bait enemies early game, but playing smartly usually doesn't involve baiting past level 12 unless you significantly fed or confident in your abilities to take someone down fast. Again, highly situational, but can be used to incapacitate or render enemy carries harmless or secure kills (Defined as 'KS', since Vayne can't be a killstealer and neither can Lee Sin =P).

Other options include:
Cleanse will enable you, combined with Flash, or just by itself IMO with
Tumble to escape Situations. You have to get yourself out of the Danger Zone since you can be killed easily if focused.

I max out Tumble, get Condemn at level 2, and Silver Bolts at level 4. The reason I max out Silver Bolts before Condemn is because I rely on condemn to stun the enemy more than the damage it does (Though, I grant you, it is moderate), and Silver Bolts will typically finish them off effectively or complete a pass at harassing. Ultimate Final Hour is an effective way to finish off opponents (Quadruple Movement Speed from passive, Attack Damage) or an effective way to flee (The stealthed Tumble throws people off just as easily as an Akali quick juke from Twilight Shroud).

General gameplay is pretty similar to most other ranged carries- last hit, don't do stupid thing (A grand example is when I was shadowing a Shaco and thinking I was a smart stalker when I used Flash and then Condemn to stun him so I could finish off. Problem was, my map awareness was so fail, I hadn't noticed MY WARDS indicating the whole enemy team was moving up in that direction. I managed to escape and secure the kill because my team had noticed and pretty much saved me as I used Tumble to GTFO and so they wouldn't see me as they passed through the bush and I passed right behind them. Use Tumble to harass or to get out of range of an enemy, but don't be an idiot like I am and roll forwards if you're just harassing. Rolling to the sides or slightly at a forward angle and not full blown charging is acceptable for harass. Smart players will catch on if you roll all the way forward and you will get smacked so hard you won't believe it. In that case that you do, Flash will often save you, but if it's on CD and so is your ultimate so you can Tumble away and dodge them a little bit, youuuuuu mighttt beee done. maybe. Your lane partner can maybe save you if he/she is attentive and not a scatterbrained turkeywing.

Farming well but also paying attention to the map so you don't miss teamfights is important. If you have 300 something minion kills but your team is dead all around you, yeah....
If your team can hold their own in a team fight (Like a Stalemate) you should be farming, pushing, or assassinating any strays. A good example of why farming instead of joining a team fight that's an assured victory or tie is because people UNDERESTIMATE the value of 300 minion kills compared to your other team's carry of 90 minion kills and 10-14 kills. A champion kill is worth roughly 10 miniion kills. If they (Say, as an example) walk right by you and are intent on chasing somebody else on your team because they think you're underfed and can't do anything, they will be sorely mistaken, as you will deck them in 2-5 shots if you are a good farmer. That'll open some eyes.

As an example, my team was fleeing a fight after pretty much they got caught in a trap. The enemy team's 3 carries were chasing them ahead of their tank, and all my team was low hp and were retreating. I come out behind them, I kill the strongest carry in 3 shots (1 second and a half maybe), kill another one, then use Youmuu's Ghostblade active combined with
Final Hour to hunt down the last one easily. Take advantage of people's stupidity. Take advantage of enemy stragglers. Pick off dumb people methodically.

Oh, and. In Teamfights. Never intiate. Never, ever initiate. Let your tank or even someone like Renekton or Sion engage because they can take a little bit of damage, and if they die, well, it's for the cause. Play smart (If you need me to define how to play smart, it's summarized in short words: Don't Die or Kill Teammates Inadvertently becasue you made them engage 5v2)

Pros:
-Extremely Adept Chaser with Youmuu's Ghostblade, Final Hour
-High sustained damage(Up to 1250 crits on enemy champions with Tumble damage bonus)
-Adept escapist with Flash and Tumble after using Ultimate for stealth.
-Able to pin down most enemies and can lockdown someone for a good 2-4 seconds with a wellaimed
Condemn

Cons:
-Squishy as my pet Sun Conure and twice as liable to be attacked than your team accounting you as a carry.
-Can easily be rendered ineffective if the enemy team focuses you or prevents you from farming well.
-What else? I can't think of anything else :P. JK, champions like Ashe and Miss Fortune can be much more effective support and carries on account of Enchanted Crystal Arrow and Bullet Time as some examples. Vayne is a single target champion, and while the enemy may go down in 4 hits, you don't have much AoE capability.