Vayne works best with the Infinity Edge or Bloodthirster as a first item for the reasons everyone has been mentioning. If I were to go all out offensive, it'd probably be Infinity Edge, Phantom Dancer, Bloodthirster, Last Whisper, and a Quicksilver Sash / Banshee's Veil. The raw AD can also do impressive things with Impale.

I am pretty new to Lol, but have played Dota style games and playing vayne i generally get the pendant with health Regen first. Then work on IE. That pendant is nice because you can make warmong's later on with it. After IE i usually get Blood Thirster if they have slower heros that you can easily catch up with or Phantom Dancer if they are fast and then the Blood thirster. For boots i know ppl like merc's treads, but i get swift boots because vayne has ok AS alrdy and she is a chaser so speed is more helpful for getting those heros that are disengaging from the 1v1 or are the only one's left after a team fight.the rest of the items really just depend on the situation that you are in and what the other sides comp consists of.

Hey thanks for all the comments guys. I've been trying out a few different things with her, and I've had great games and some stinkers. I actually just played one of the most epic games of LoL I've ever had. I was Vayne on a team with an Anivia, Teemo, Yi, and Chogoth, and we were against I believe Jarvan, Sivir, Zilean, Warwick, and maybe it was Kassadin. I've been finding the biggest problem with Vayne is that it's tough to avoid feeding in the early game, since she's so squishy and really doesn't do a whole lot until she has some items. This game I was dying a lot early, but I was at least staying even (basically trading myself for Jarvan or Sivir, who I was laning against with Teemo). Now you can imagine my suprise when I realized that me breaking even was the best record on our team, and that Yi and Chogoth had no kills and had fed Zilean up to level 17 by the time I got to like 13.

Now, I mostly queue with a group of m friends, and they are all level 30s, but I'm only at 22, so basically I'm used to getting my ass kicked all the time, and I rarely find myself in situations where I have to be the one to take charge. This was just me by myself, so apparently it falls to me. Long story short, we were on our last legs (I even typed at one point that it might be time to surrender, but didn't actually start the vote), but we had still held them at bay for long enough (somehow), for me to get IE, Black Cleaver, and Phantom Dancer, at which point I could kill just about anything by glancing at it. Final build was Merc Treads, IE, BC, PD, Banshees Veil, and a BFSword that I was going to make into a Bloodthirster but we won the game before that was really an option. I ended up 25-17-6. Our team had a grand total of 41 kills, so I accounted for more than half.

Then of course the next game I played with her I went like 2-15-4, so take that how you will.

As much as I find AD is the most important stats for Vayne, if you want to build her solely from her burst and Tumble, you can consider building Trinity Force to upgrade your Tumble burst damage.

One thing that people find it hard when trying to kill Vayne is if Vayne keeps tumbling everywhere avoiding stuffs and counterattack. This is especially more deadly if she has Ult up since you stealthed and can reposition elsewhere, making any foe u attack take a second delay to attack you again.

But some people will disagree since one of it's core item (Sheen) scale with AP. While Vayne doesn't have any skill that scale with AP, that item basically pointless. I say it's only pointless if you only play her right-click-and-kill only without playing tumble all the time. Also, Sheen + Tumble (and the lucky 3rd hit from Silver Bolt) = Tons of damage. Upgrade to Trinity and it's even stronger. Combine with the critical rate of PD and IE and you got yourself a scary OP Vayne that will do 1k+ damage out of her Tumble alone.

My 3 cents

So I tried a few games with an AD/Crit Vayne and while I have to admit that I was hitting harder per shot I wasn't as successful. Could have been my team mates (you never know with solo queue), but honestly I think what I was missing was all the defense that's in my build. With merc treds, wit's end and madred you have tenacity, 50 MR, and 25 armor. It's also decent DPS that really helps with Vayne laning.

Perhaps better players don't need those crutches and can build pure AD/Crit, but as a newbie still learning the ways of Vayne (Q+E is great for harassment and making sure they can't hit back!) the extra defense really helps.

I get cleaver first for some reason, I just tend to enjoy the attack speed and armor shred earlier

health regen
boots
vampiric scepter
black cleaver
bloodthirster
phantom dancer
infinity edge
great for kills

Depends for me, I usually build against the other team.

3 squishy I'll go AD/Crit:
Start boots and 3 health
Boots > merc treads / greaves depending on what you need
BF sword
Zeal
Black Cleaver
Bloodthirster (Interchangable here, I usually go Bloodthirster first because solo ques don't run supports often)
Zeal > PD
IE
Banshee's

2-3 Tanky DPS :
Start cloth and 5 health pots
Usually Greaves
Cloth > Madreds Bloodrazor (get recurve bow component first)
Wit's End against 2 AP, otherwise SotD
Frozen Mallet
Then build more attack speed for more damage OR more defense for better survivability. Starks helps as well, attack speed and a little lifesteal. Level up silver bolts first, and if you harass in-lane make sure you can get three hits and get away safely.

Hope that helps.