Hello everyone! I am a mid-high Silver player of absolutely no renown that loves Vi and has used her and several other champions to consistently carry myself out of Bronze across several Seasons. Getting to Gold is my goal for Season 6 and if I stopped getting drunk and playing in Silver 1 it might happen!
This guide is only for Jungling Vi as that is her most efficient role other than niche picks in the top lane that are more difficult to make work in ranked play.

Vi is foremost an AD bruiser/offtank that is loads of fun to play. The champion has a colorful, no-nonsense attitude and her kit, while simple, has a great amount of damage, burst and utility. She is easy to play, but difficult to completely master. A well placed ultimate combo can disrupt the entire enemy, chunk the enemy carry and allow you to escape while your team cleans up behind you. She can split-push well and take towers acceptably fast with her AA reset on E and easily escape most other champions if they decide to chase. She excels at hit and run and dueling most other champions at all stages of the game.

Pros

Strong duelist
Very mobile and can cross terrain
Can solo dragon early
Can delete squishies
Good mix of tankiness and damage
Teamfight disruptor
Easy to catch fleeing enemies
Flash+Q Ganks are very effective
Effective counter jungler
Good clear speed and aoe ability for small monster clearing
Scales very hard with AD

Cons

Very difficult to catch up if behind
Predictable if flash is down and the enemy wards well
Bursted down in teamfights even if you do your job well
If you're not careful, E can reveal your location in the jungle to enemies
If you miss Q, you lose a lot of damage, can leave yourself vulnerable and can be kited

I always start the camp closest to bottom lane in order to maximize the efficiency of your leash, but I prefer to start Red Buff as you shouldn't burn through all your mana early enough to warrant Blue Buff first. If the ADC and Support do their job correctly, you should leave the first camp with almost full health and no potions used. Always smite the first camp. Gromp gives a DOT to attackers and Krugs give a stun every 6th AA that will save you some more health. Take W at level 1 and E and level 2. Always make sure to E back into your own jungle within your point of view so that the shockwave doesn't give away your position to a wandering enemy.

After the first camp, go straight to the second monster camp, not your buff. Pop a ward out in a bush where a counter-jungler is most likely to pass by. This will ensure that you don't get cheesed at level 2. After that, kill the Golems or Brambleback. Smite should be back up by now, so I always use it on the Blue or Red Buffs. If you started Red, immediately go gank Mid or Top/Bot. Put your first point into Q at level 3. This early gank ~3 minutes into the game almost always gets First Blood for your team or burns several enemy summoners spells. If you started Blue, go take the Rift Scuttler first instead with AAs to get some health back and be looking to gank right afterwards.

After you gank and provided you have enough health left, go straight to your second buff and take it. A ward should be back up by now, so if you don't know where the enemy jungler is, I would suggest throwing it down while you engage the buff. Clear the rest of that side of the river and you should have enough money for your first Recall.

Once Vi hits level 4 and you have a Hunter's Potion, you can solo dragon provided you have full health and mana. Remember to ward the area and ask for assistance anyway if possible to minimize the risk. If the enemy jungler or laners wander by, Q over the dragon pit wall to escape Do not use Q on dragon, save it to either escape or engage if a favorable fight breaks out over dragon. Keep a close eye on the minimap during this encounter to keep yourself from getting caught in a bad situation. You can easily kill every dragon except for the Elder by yourself for the remainder of the game. Watch the minimap and use this to your advantage. Ocean and Fire Drakes are the two most important and should be stolen if at all possible even if it means dying to the enemy. I personally never fight over Mountain or Cloud Drakes unless I am 100% certain that I can take it and get away.

You should be constantly farming and ganking lanes during this time provided you have over 50% of your health. Clear a camp or two and look for oppurtunities to help your team out. With Tracker's and Oracles, you should be warding your teams' blind spots as you travel around and clearing enemy wards to facilitate ganks or roams.

You should hit level 9 around 12 minutes into the game. At this time with full health and assistance from your top laner, you should be able to kill the Rift Herald. Most top laners will want to take the buff, but I feel it is better suited for Vi especially as it really increases the burst from your ganks and recharges very quickly considering that you should be roaming the map.

Ganking early to mid game always follows the same pattern. If you have ult and the enemy champion doesn't see you and is in range, ult on to them and after the dunk quickly short-charge Q back into them. This will do a good amount of burst damage and apply two separate instances of CC on them during which they are unable to retaliate. Remember, your ultimate cannot be stopped by anything during the dash and always applies its damage. There are certain champions that can abuse the small window between your dash and the dunk. Lee Sin, for example, can kick you back at the end of the dash during the dunk, but the damage is always applied. Always.

If you do not have your ultimate yet or it is on cooldown, you can still gank provided you land your Q on the enemy champion. This is sometimes difficult to do as the range of the dash isn't quite enough to make it to the middle of the lane from the bushes. This requires you to line it up and run a short distance into the lane while charging it before unleashing it upon your target. Flash makes this so much easier since it effectively doubles the range of your dash and adds a layer of deception that will confuse your opponent. There are three interactions between Flash and Vi's Q: You can Flash during the windup of Q, finish the charge, and unleash it. You can also, flash and short-charge Q into somebody and finally, you can flash as you release your Q. The last one is the most difficult to do as it requires good aim and good timing, but if you pull it off it is the smoothest and fastest of the three. Keep in mind, you can aim your Q in a different direction and then Flash towards the enemy champion and release it to add to their confusion. Vi's Q isn't hard to dodge, but a good Flash>Q almost always connects.

One of Vi's main combos is: R>AA>E>Q>AA>E
This combo will delete almost any squishy while locking them down between the raw burst and two procs of Denting Blows. Keep in mind that Vi's E is an auto-attack rest, so as soon as Vi's fist connects with your target during the attack animation, press E to reset the animation and double-attack while charging the base damage of your punch with E's blastwave (and a spellblade proc if you took Iceborn Gauntlet or Trinity Force)

Another good combo if you're worried about the target fleeing or need to buy some time for your team to clean up is: Q>AA>E>R>AA>E
If you hit the Q, you can generally get in an AA and an E. The enemy will either try to fight you or burn summoners to get away. Since the range on your R is quite large, they will usually try to Flash and Dash away from you, but unfortunately for them they won't get away from your R with just one or the other. Ult onto them as they flee, land finish the combo, and by that point if they're still alive, Q should be off cooldown which will further lock them down. This is devastating with the Phage passive and Tier 2 boots as it will allow you to easily stick to your target. With an Iceborn on top of that, only the fastest, tankiest champions will survive or get away.

The last combo of any note is: R>Q>AA>E>AA>E or Q>R>AA>E>AA>E
This is meant to lock down an enemy champion for as long as possible in order to buy time for your teammates. If you manage to finish it, the combo will still apply all your damage, but the up-front lockdown is what separates this combo from the others.

The utility of this ability is often underrated. Most people just see it as an auto-attack reset and a damage increase, but the range on it is quite impressive allowing for several interesting interactions. First, it increases the reach of Vi's next auto-attack which means you don't have to be quite as close to connect the blow on a fleeing opponent. Second, it sends out a blastwave in a cone from the point of impact. If someone is running away with low health and out of range of even your extended auto-attack, hit a minion in line with them if possible. This will proc the blastwave which could very well get you a kill you wouldn't have gotten otherwise. It also works well for getting an assist on a dying champion you can't make it to in time or stealing a kill out from under someone (I do not endorse doing that, but it is still something that is possible for Vi).