My name is Quadda and this is a Guide for Shyvana.
Shyvana is a tanky jungler with high damage, high mobility and the ability to carry.

Pros:

+ High mobility.
+ Great map control.
+ Can carry a game.

Cons:

- Does not have hard CC.
- Hard to gank.
- Not very tanky until 6.

For runes I use:
x9 Attack Speed Reds.
x9 Flat Armor Yellows.
x9 Scaling Magic Resist Blues.
x3 Movement Speed Quints.

The reason for this is that I feel that x9 AD Reds on her are not useful because her scaling isn't great with AD and with more AS you get to proc Q more often since it lowers the cooldown with each basic attack, also helps clearing the jungle faster for the FF stacks.
Flat armor yellows for early armor.
Scaling magic resist since you'll build Banshee's Veil as your fifth item. Again, against a double AP comp you could always buy an early Hexdrinker.
Movement quints are really great since it improves her already high mobility.

You'll start with Hunter's Machete and Health Potion as most junglers.

In the first back, you can buy Madred's Razors, Boots and a Stealth Ward if you have enough gold, the Stealth Ward will help with your counterjungling, as well as help your lanes with wards to prevent ganks.

Feral Flare is a great item as it gives attack speed, attack damage and each of your basic attacks deal bonus magic damage, make you clear jungle camps faster and it'll give you a ward

Blade of the Ruined King it's a must on most if not all games, gives attack speed along a nice AD and lifesteal, makes you duel even better, gives you a slow and more damage per hit.

For boots you can choose either Ninja Tabi or Mercury's Treads depending on the opponent team.

Then you can build Banshee's Veil or Sunfire Aegis

If they have a fed AA reliant champion, such as Tryndamere, Vayne, Yi etc. You can buy a Thornmail

If you're getting fed early you can get Trinity Force it gives you really nice stats and along Blade of the Ruined King gives you a great carry potential

Sequence is R>W>E>Q since Q has bad scaling compared to W and E, besides you already reduce it's CD with basic attacks, so you should max it last. W gives you nice mobility and E gives you a MaxHP% damage. R Doubles your passive's bonus armor and magic resist.

Passive:

Gives you bonus armor and magic resist that are doubled when in Dragon form.

(Q) Twin Bite:

Twin Bite Makes your next auto-attack hit twice dealing a % of her AD on the second hit.

(W) Burnout:

Burnout Makes you deal AoE magic damage while giving you movement speed. The duration can be extended with basic attacks.

(E) Flame Breath:

Flame Breath Marks the enemy while dealing magic damage. Basic attacks against the marked enemy deals bonus magic damage equal to 2% of the target's max health.

(R) Dragon's Descent:

Dragon's Descent Makes you dash to a target location knocking back enemies in her path, also doubles her passive bonus armor and magic resist.

Early Game:

You'll want to complete your Feral Flare as soon as possible, so gank only when it's guaranteed the kill or when the enemy laner has no flash. Push the wave only to deny cs and exp from your opponent, as it delays your Feral Flare. Also look to secure dragons, you can do drag when your enemy bot lane is dead or the enemy mid is dead and your top has TP, place a pink ward whenever you're about to start it.

Mid Game:

At this point you should have FF completed and a Blade of the Ruined King, so you'll want to start pushing towers and keep control over Dragon and Baron.

Late Game:

Shyvana is really great in late game as not only you'll be tanky, but you'll also have great damage, but you won't offer much cc, so you'll want to try and kill their carries, while peeling for yours.