In my opinion, there are only 2 effective ways to play Vi. The first one (and more compelling for me) is the offensive approach: put 21 points in offense to destroy enemies fast and thorough, getting some basic defense as well.
The second one will be explained below.

Tier 1
Summoner's Wrath: 5 additional AD and AP on your [summoners/ignite.png] respectively reduced MR and armor on your [summoners/exhaust.png] should add a slight edge that may make a difference in some cases.
Fury: to get your [skills/vi/w.png] rolling and basically improving your sustained damage.
Sorcery: more spells mean more damage and more [skills/vi/p.png].

Tier 2
Deadliness: because Vi scales so good with AD and to get to past tiers.

Tier 3
Weapon Expertise: pretty obvious. All of your damage is physical so 8% armor penetration will help you apply that.
Havoc: percentage damage is a pretty obvious choice as well. Best choice to get to past tiers.

Tier 4
Brute Force simply adds more dps on both styles and auto-attacks.

Tier 5
Vi does not make great use out of critical strikes, hence we skip 10% critical damage (Lethality) and the AS buff after crits (frenzy). The AP masteries won't help that much since only [skills/vi/e.png] has an AP scaling on it.
Butcher won't pay off except for laning and pushing and putting points in there sacrifices teamfight potential and damage versus champions.

Tier 6
The reason why we are putting so many points in that tree: about 2-3% more overall damage on your targets, especially in the crucial phase when someone gets low. Crucial choice in my eyes.

There are a few runes that might be possible to run, my favorite is always highlighted, my second choices are like that.

Marks and Quints:

AD is plain effective due to her great scaling in almost every aspect. Every spell benefits from it and auto-attacks deal more damage. Best scaling.
Attack Speed increases your damage via [skills/vi/w.png] and auto-attacks, but at the cost of extra damage on [skills/vi/q.png][skills/vi/e.png][skills/vi/r.png].
armor penetration should be inferior due to Vi's great scaling, but in some extreme cases it might be worth taking some extra armor off your enemy. Consider maxing [skills/vi/w.png] in this case, since it is % damage and does not depend on AD that much.


Quints:

life steal can help you sustain in your lane in case you get counterpicked and/or having a hard time to farm. It costs you some decent early game damage though, but these runes get even better over the course of the game.


Seals:

usual choice should be armor: provide the highest effect on the field, making it easier to tank minions/turrets early on and especially trading with your lane. Get them when your lane deals mostly physical damage.
HP/level in case your lane and/or whole enemy team lacks AD damage but also % damage.


Glyphs:

in most cases you should get magic resistance / level. They surpass the flat ones at level 9, and since you are on a solo lane, they pay off pretty fast.
In some extreme matchups it might be worth getting flat MR runes instead to help your laning and/or enemy jungle ganks. However, those occasions should be rare.
Enemies really lack AP damage: get attack speed to help your [skills/vi/w.png] stacks to apply earlier.