While it's true that Vi is a not the strongest jungler, she is FAR from the weakest, and is a TON of fun to play. She is hyper-aggressive and has some pretty gnarly ganks. Put Vi in the jungle when you have someone who lanes top better, but still want to run Vi for her abilities and team synergy.

The first thing to understand is that Vi is not an initiator, she's an isolater and disrupter. this means that, should you play Vi in the jungle, you need to have a strong initiate.

Vi jungles best with characters that have strong CC. This lets someone else lock them down while you chain into her very effective CC maneuvers.

The best team for Vi is one that locks characters down, and lets her combo her CC for long chains of immobility. She also favors champions that are attack-damage casters, as her denting blows shred armor.

Best Top Lane:
Malphite, Warwick, Renekton, and Darius.

Best Mids:
Malzahaar, Swain, Zyra, Ryze

Best Support:
Taric, Sona, Blitzcrank, Leona

Best Bottom-lane DPS:
Vayne, Ashe, Varus, Draven

There are some non-traditional picks that work REALLY well with her, as well:
Twisted Fate(AP)/Blitzcrank in the bottom lane with Corki (AD) in the midlane does serious work.

Wolves->Blue is a must. From here, you'll want to lane gank directly after. You want to gank the lane that is next to your blue (Top/mid for blue-side, bottom/mid for purple side). Take no longer than your wolves. Once your wolves spawn, abort the gank. Vi is more DPS-orientated early, and not very tanky, so play the gank safe. If you cannot land a good gank, return to wolves.

Wolves->Wraiths->Red is your next path. You'll want to attempt a gank once you have your red buff. Try and clip someone on "that side" if you can as well (Top/Mid if purple side, Bottom/Mid if blue.) If you go to Top/Bottom (respectively), grab golems and sweep around for a re-gank. If you ganked Mid, sweep into Wraiths/Wolves, and attempt a gank, too.

This is your general pattern to follow. If you attempt a gank top/bottom (blue/purple respectively) grab golems afterward. If you gank mid, grab wolves->wraiths. If you're on blue-side, try and gank the nearest lane. The goal is to keep your camps on cooldown as much as possible, while also having as much lane presence.

This is when Vi really gains her legs. Once you have your 4 core mid-game items (Phage, Hex Drinker, Vampiric Scepter, Spirit Stone) and Tier-2 boots, she escalates from being a semi-fragile DPSer to being a strong CC/Damage machine with reasonable survivability.

The goal in this phase is to continuously wage asymmetric warfare. With the help of your movespeed quints, you will have 413 movespeed. This will move you around the map at a fairly fast clip (Especially if you use her Vault Breaker ability to wall jump and time it to be available once you arrive in lane.)

An example of a how engages should go:
Blue-side Team: Swain(M) Malphite(T) Vi(J)
Engage: enemy is at mid-field mark. Swain pushes his lane to just before the mid turret (bonus if mid is down). Swain and Vi will go top after getting Swain his Blue Buff. Swain will initiate with a Nevermove circle. Vi will combo in with her Vault Breaker. Malphite will finish with his Unstoppable Force. If the enemy has not died yet, follow with an Assault and Battery, Vi's ultimate, into another Swain Nevermove. The massive CC should completely disable the enemy champion, and the burst should eliminate even the tankiest of toplaners quite effectively.

These types of heavy-CC asymmetric skirmishes are why you put Vi in the jungle to begin with.

This is where having an initiator will really start to pay dividends for Vi. At this point you should have your two major tanking items (Frozen Mallet and Spirit of the Ancient Golem) And have some good progress toward your next major damage item (Maw of Malmortius). If the game is going heavily in your favor, you may even have your 4-item core built (Maw of Malmortius, Bloodthirster, Frozen Mallet and Spirit of the Ancient Golem.) Though Vi starts out somewhat fragile, her base stats let her get very tanky by end-game. To this effect, her role in team-fights is to separate out any ranged DPS.

An example of teamfighting:
Scenario: Midlane at your Outer Turret.
Team: Malphite (T), Vi (J)
The teamfight begins with Malphite landing an Unstoppable Force on a cluster of enemies. Vi will use her Assault and Battery ultimate to cut through the enemy team (hopefully knocking up a few people on her way to the back ranks) and stick to the highest-priority damage target, where she will combo this into her Vault Breaker ability. She will be facing toward the body of teammates (Her ultimate puts her BEHIND the enemy it is cast on.) She will lace her Excessive Force blows between auto-attacks. This should have all of her abilities on cooldown (the vault breaker has a fairly long cooldown). You will now run around and apply Frozen Mallet slows to escaping enemies, allowing your DPS to break down the targets. Once primary target objectives are eliminated, Vi needs to find high-armor targets and get her Denting Blows applied as soon as possible. This will allow the highest amount of physical damage to penetrate their armor, thanks to Denting Blow's 20% armor reduction. Use Excessive Force as it comes off cooldown to lace extra attacks in and build to a 3rd blow faster.

The essential gist of her team play strategy is to cut through front ranks with her ultimate and separate out high-damage enemies from the group to allow your team mates to eliminate a focus-fire targets.