This is my build for Shyvana. Hope u like it & use it :).

Shyvana, The Half-Dragon

There are those few dragons in Runeterra who have mastered the intense magical energies that course through their unique anatomies, evolving into an elusive Celestial Dragon. These powerful and enigmatic creatures spend most of their time hidden away from the lands of men. However, there are those who find themselves drawn to civilization and who take on human forms to immerse themselves in the world of humans. Occasionally, one finds itself drawn to the spirit of a human and they have relations. Sometimes, in the slimmest of odds, the union will bear fruit and that offspring will be one of the few half-dragons to have graced Runeterra. Shyvana is one such creature, born to a simple Demacian farm-girl and her Celestial Dragon father. Left to be raised by her mother, her oddities were something that couldn't be hidden from her community, though she was protected by her family. Once her family had perished in disaster, she was left alone for the first time - truly alone. Shyvana quickly became an outsider and it wasn't long before the community's fear began to put her in danger. Facing mortal danger at the business end of an angry mob, she was rescued by the traveling noble, Jarvan Lightshield. The noble was a man who strangely knew exactly what she was and took her under his wing. It wasn't long before she was offered a chance to use her natural skills in the service of Demacia, as Jarvan initiated her as one of the elite guard. Following Jarvan into the League of Legends was only natural.

Shyvana is a fierce warrior, with the blood of one of the most powerful magical predators in all creation flowing through her veins. Unlike some of her fellows, she is a subdued personality - somber, cool, and collected. However, when the moment strikes, her draconian heritage manifests and there are few that can stare her in the eyes and not feel the primal urge to flee.

''By the blood of my father, I will end you!''
- Shyvana
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Pros:

- Great mobility
- High damage in any game phase
- Lots of AoE, specially in Dragon Form
- No mana, so you can use abilities as much as you want :)

Cons:

- Not a strong ganker
- Hard to master
- Takes time to get used to.

Mark:
Greater Mark of Attack Damage Gives you early advantage in early-game.

Seal:
Greater Seal of Armor Gives you more armor, so u take less physical damage.

Glyph:
Greater Rune of Scaling Magic Resist Gives you more magic resist, so you take less damage from spells.

Quintessence:
Greater Quintessence of Attack Damage Gives you advantage in early-game & mid-game.

- 1/1 (Gives your spells some boost.)

- 4/4 (You have more attack speed.)

- 4/4 (Increases your attack damage.)

- 4/4 (You do more damage to turrets.)

- 3/3 (Increases damage dealt by you.)

- 1/1 (Gives your more armor penetration.)

- 2/2 (Your critical strikes deal 5% more damage.)

- 2/2 (Increases your attack damage.)

- 1/1 (After doing a critical strike, you gain 10% attack speed for 2 seconds.)

- 3/3 (You gain more armor penertation.)

- 1/1 (Aou deal more damage to low health targets.)

These are the items u need to buy:
Doran's Shield - Gives you armor, health & health regeneration(sell this before buying Infinity Edge).

Ionian Boots of Lucidity - Gives you movement speed & cooldown reduction.

Frozen Mallet - Gives you health, attack damage & CC (Crowd Control).

Enchantment-Alcarity - Gives you even more movement speed.

Wit's End - Gives you attack speed & magic resist and also has a great passive.

Sunfire Cape - Gives you armor, health and has a damage-over-time passive.

Maw of Malmortius - Gives you attack damage, magic resist and has two awsome passives.

Infinity Edge - Gives you attack damage, critical strike chance and increases your critical strike damage.