This is something I can help with as i am a jungle main and a Vi main :)
I tend to start blue, even though I know of the new red>krugs route, I just enjoy having mana and having plentiful amounts of it.
Now if you start blue, you won't get level 3 until you get murk wolves, if you start red, you'll get level 3 from red and krugs. Vi's clears are extremely fast and fluent. They're very safe. I always E on the final hit of her passive W, and always start W. Max Q, then Max E, but some max W second. Up to you.
Now if you have talisman and machete, with one smite up, and level 3, you can solo dragon but it's a pain to do at that early but is possible as i've done it in the past. I go for dragon at level 5 and when I have two long-swords for a bit of damage, but feel free to try to go earlier but at a higher risk of dying taking it.
Ganks are a different story. I gank at level 3, either from red or after murk wolves or scuttle, but I do gank very early. Ping for your laner to be ready and use your Q to initiate. Best thing to do is Q them in to your laner since the Q as a knockback. When you're ganking, always E with the final hit of your W passive for the most damage. Vi is capable of skirmishing in solo battles and can pick up solo kills depending on where you in terms of build and level. I either use my ultimate as utility to knock up, or to secure kills. Either way, it works extremely well in both of those aspects, as it does damage and prevents the enemy from getting way from the gank. Ganks with Vi are quick and fluent as she can roam fairly quickly and gank each lane.
Invading the enemy jungle works very well on Vi as well as her damage with her W is strong. If you catch the other jungler depending on who they are, you may be able to 1v1 depending on your build/level/hp.
For masteries, I can't remember them all off the back of my head, but make sure to run CoTC, as it gives you shield from your Q and allows you sustain more damage and stay in the fight.
The build is simple, I go jungle item with the 60+ attack damage>MR boots/AR boots>Triforce>Dead mans/Spirit>Steraks Gage/Frozen Mallet, for the final item, i see a lot of GA but I go another damage item as I personally dislike guardian angel as an item.