Shyvana is best used as a attack-speed crit chance DPS. Since her passive enhances her abilities based on her basic attacks attack-speed is very important. However you can't do a lot of damage with just attack-speed. That's where crits come in. a phantom dancer will give you both a lot of attack-speed and crit chance. A Trinity Force is a must early in the game. If you have not noticed Shyvana's Twin Bite ability at level 5 does about the same damage as three basic attacks. But, with a Trinity Force the first attack will do 150% increased damage, making Twin Bite a deadly ability with a short cooldown. Leveling twin bite to level 5 first is not suggested unless going all damage.

Use flame breath to start to lower their armor, burnout to catch up, and twin bite and basic attack to finish. Ult if needed.

P.S. This build works for my play style, check out other builds if you do not find success with this play style. Thank you for reading! Good luck and have fun.

Pros:
-Great Gap closer
-No mana, all cooldowns
-Lowers enemy armor well
-Super high damage

Cons:
-No CC
-Hard to master
-Requires good laning partner