Strategy and Tactics

Strategy

Early Game

Vi is an incredibly potent brawler throughout the laning phase. She tends to do much better against melee champions top lane, so ideally you would want to pick her into something that isn't Vlad/Teemo/Chogath. Ranged Champions have the leisure of getting free harass on her 1-6 and due to the long wind up time of Vault breaker, people can often times just walk away when they see you charging it unless you are in a brush (which is a tactic you should try to take advantage of. A fully charged Vault Breaker out of a brush can clear a large gap and can often times surprise your laning opponent). Since we will be maxing E you will want to use that as your primary form of harassment, however be careful not to use it excessively, as it does tend to push the lane regardless of how you use it, so make sure you have proper ward coverage.

If you feel relatively safe (know where jungle will go) and have placed your ward you started with, you can start chunking your enemy with carefully placed E's either on them, or even on the front melee creeps since the AOE cone stretches very far back to wither down your enemies health. Your Q is both your engage and disengage tool, so I tend to play it safer and reserve using my Q unless I am gauranteed a favorable trade, or I see or sense a Jungle gank coming, at which point you can simply Q towards your tower and escape from most situations.

Post Level 6 Laning

At this point, you should have already decided how you want to play the lane out. If you are very fed and are doing extremely well, you might have a Brutalizer, or at least both Long Swords and boots at this point. Vi can do incredible damage with her full combo combined with additional E's to pretty much any champion. Try to use your ultimate at an opportune moment to disadvantage the enemy as much as possible, for example if you are laning against a Rumble or Riven, use it to dodge their ultimate and engage them for maximum effect. Vi is very strong after she gets 3 points in E so take advantage of this fact. Some matchups may be difficult so you may have to play safe until you get a few free E exchanges off on them to chunk them low enough. Often times you may be able to blow the enemys escapes or Flash cool downs with a well timed Q, which leaves them completely vulnerable to a free battering with your ultimate combo. In general use your head but try to get a feel for how much damage you can deal to the enemy and make sure to always get your third hit of W off whenever possible, otherwise you will end up losing most trades to most top laners as the steroid and increased damage is too much to pass up on.  

Tactics

Roaming to Mid

Once you have acquired your ultimate, you should utilize Vi's incredible roaming potential to the mid lane to try to make as much as an impact as possible. Due to the fact that most of your trades will innately push up your lane, the enemy team will think nothing of you constantly pushing out your lane to the enemy top laners tower, so you should take advantage of this certain scenario and as soon as you are finished trading, push out the lane and attempt to roam mid. Due to Vi's incredible gap closing potential with a blind fully charged Q out of the brush, if the enemy mid laner has not warded, you can guarantee kills on every single one of your ultimate cool downs (or at the very least blown summoner spells). This constant pressure will allow your team to apply more pressure throughout the map and help shut down the enemy team.

Team Fights

In general, Vi will play the role of the initiator / front line. Her skill set makes her incredibly durable as well as very good at killing tanks (due to the ridiculous armor shred from W + BC) so having her peel for your AD carry is a very viable playstyle for Vi. Additionally, due to the diving nature of her ultimate, you can also utilize her to disable any enemy champion and easily take them out of the fight. Try to reserve your ultimate for their squishy AD/AP carry but use it at your discretion when the situation calls for a good knock up. Once you acquire a Frozen mallet, assuming your AD carry is very fed, you should sit on anyone that tries to damage them and simply peel them off with your Q+R. Vi can do very high damage to pretty much anyone so don't be afraid to hit a tank (assuming they are the best available target).

Tips and Tricks

Most walls can be cleared by charing up a full Vault Breaker and dashing through them.

In laning phase and team fights, utilize the AOE Cone that is behind your E, often times you may be able to poke someone out at max range by E'ing a melee creep in lane.

Assault and Battery(R) can be used to dash to enemies champions through walls.

You can close a distance of a fleeing champion by Q → Flash → Q → R

Assault and Battery (R) will place Vi behind your target. You can follow up with a quick Vault Breaker (Q) to knock them back into your teammates.

Boots of swiftness is very useful on Vi. Because it reduces the slow on Vault Breaker(Q) by 25%. Plus Relentless from the mastery tree can stack this effect. Making you move faster in Vault Breaker(Q) and being able to reduce the enemies time to react to your assault.

Look at her left cheek. Her name is written there. That is all.

Synergies and Counters

Synergies

Champions that combine well with her dive potential can do very well alongside Vi in team fights. Having an Orianna balled Vi charge into an enemy team and get a wombo combo off on everyone can seal many team fights. Similarly, having a Vi jump into your team while Lulu ulted will disable everyone heavily and due to her innate tankiness, your team will be allowed to clean up and deal damage for free while they attempt (unsuccessfully) to peel Vi off their carries.

Counters

 In general Vi does not lane as successfully against things that can harass her at a range and have room to dodge her Q, such as Chogath, Vladimir, Teemo, etc. However, Chogath and Teemo are especially pesky in lane against Vi. Chogath's natural sustain and tankiness makes it hard for Vi to lane well against him, as well as the fact that his silence will cancel your Q mid charge. Teemo's early game is just very hard to deal with as most melee champions, and as soon as you begin to charge your Q he will move quick and run away from you so it is impossible to land it on him unless you get on him from the brush. Also when he hits 6 he will be able to place a shroom near him and every time you charge up a Q or ultimate him, you will be hit by the pain of a thousand Teemo poisons. 

Irelia is a special case and she just lanes well against most physical damage dealing top bruisers such as Riven, Zed, Lee Sin because her true damage allows her to build armor and still out trade these champions. Vi is no exception to this rule and while she can potentially win the lane, it is heavily in Irelia's favor.