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

Shyvana still new in league of legends but really interesting champ so I had some games and started the build. Now you have to understand that its not final build just first steps and is suited according to my play stile. So pls don't judge me hard and sorry for my bad english :D

Pros

-Good AOE dmg
-Good Chaser
-Fast jungler if you go that way
-Can take some dmg
-Great escape and caching mechanism

Cons

-Lack CC
-if build pure carry early game could be hard
-dead if focused out of her dragon form

Well nothing really to explain here I tried using arp for marks but as I lvl Burnout which does magic dmg so its not that effective. So I decided to go AS for marks I might also consider going AS on quint but don't have all 3 at the moment. I saw many builds going AS for seals and glyphs imo some resisit are better for her but its just mine point of view

As I don't like going full on 1 tree I spread my point accordingly and I think its self explanatory. Some resist from def tree. Awareness for better lvl speed.

Edit* Tried going def tree and results were bad as well as utility tree is useless for her as she doesnt benefit from cdr or mana regen and the only mastery worth taking is Awareness

well its still pain in the **** but I max Burnout first followed by Flame Breath and Twin Bite last but considering to switch and max Twin Bite second
Edit* Did some testing and and Twin Bite is the 2'nd to max and Flame Breath the last as the armor reduction bonus remains the same from 1-5 lvl and as we dont focus on ap you cant really use it as a nuke.

As for skill usage in fight I think E-W-Q-Q-R suits her well. I use her ulty Dragon's Descent to finish fleeing enemy's mainly but in some situations it serves me as a perfect escape mechanism as you can jump over the walls you can also use it to get some dist between you and your enemy giving some extra dmg while repositioning your self for a kill. Walk past enemy and basic atrack pop Burnout and Dragon's Descent over him turn back Flame Breath now as he got some dmg from your ulty and burnouts trail you can pop Burnout again and finish him of use Exhaust or Flash if needed to secure your kill

As attack speed gives bonuses of her passive I give her quite much of that. Now I think that Wit's End is a must for her yet you might consider Malady as for Sunfire Aegis well its optional I like some extra health and armor plus some dmg per sec so I take it you might skip it and rush strait to the Bloodthirster now for the last items you can pick either Infinity Edge as I do or take Trinity Force . Last I pick Phantom Dancer but you might consider taking Madred's Bloodrazor as its the late game and they have plenty of hp. I'm still thinking about Hextech Gunblade as you get nice life steal and some spell vamp but on the other hand you lose AD. After some testing I decided not to take Hextech Gunblade as ability power is only useful for Flame Breath and Dragon's Descent as for spell vamp you wont bee spamming these two that often to benefit from it. The only skill you might consider getting it for is Burnout yet I dont think it does enough dmg to benefit from spell vamp its the same thing as with Singed's Poison Trail
I also saw some ppl going for The Brutalizer couse your cd are short so you don't really need cdr and +15 good but not enough to fell the big difference as for Youmuu's Ghostblade the only stats you want would be 15 % crit chance and thats all +20 arp meh not worth 15% cooldown useless as for its active well its good but I think you have enough speed from Phantom Dancer and Burnout other champs. More over your ulty doesn't have cd and the only thing that counts is fury which you get more as you basic attack. You have more atack speed = you can spam dragon form more often = you can take and do more dmg.

Summary

for boots

or

2'nd item

or
also I think would be a good choice for early mid game and later on you can switch with either Phantom Dancer or Madred's Bloodrazor

now I like to get some extra hp

or and build it into Now its your choice I take Sunfire as I play only solo 5x5 which means I cant fully rely on my team mates as they might suck and not always do their duty as they suppose to. Yet if you play premade you can go with Trinity if the game lasts and tank can serve its purpose I often sell Sunfire and get Trinity as for with crit chance over 70 % and its passive you can back door like a boss and take down the towers or inhibs in seconds.

next life steals and dmg(as you basic attack you cant miss on these)

or I prefer 1 option

for last slots

or 2'nd
or

As you see I have included some Ap items as options yet Shyvana wont get good benefits as other hybrid champs.
EDIT I took away as an option because it is just not for Shyvana AS boost and 30 ad is just too small boost for price you have to pay and as I said before Shyvana is not a hybrid. Her E is too weak to relay on as a nuke, besides you miss some times. Her R does not benefit enough from AP either so why waste the slot. I also considering taking out Hextech Gunblade 4 out of 5 still makes it viable option as you can benefit from all its stats except AP yet spell and life steal some ad and its active makes it a good item. though not really effective in 1vs1 situation as to bring its full potential you need to give huge amounts of damage to creeps or jungle monsters to benefit from the spell vamp and only using Burnout basically if you run into creep wave pop burnout and start hitting you might be full in a few secs. In fact such strategy pulled me some kills let say you have like 30% hp left and enemy chases you in a minion wave, pop burnout dragon form on them E and Q Q as you can the amount of hp you regenerate is surprisingly good actually I walked with a kill and a full hp bar in some situations yet dont forget not so effective in 1vs1

I also come up with a optional early build

with this you have enough resist for now and a good dmg output plus the free wards. After that its up to you what to build some hp would be nice of course. So get sunfire or Trinity as you wish and dont forget to turn Wriggle's Lantern into Bloodthirster ASAP

farming is easy just run into wave pop Burnout and basic attack

Shyvana could be a really fast and viable jungler as her AOE dmg lets her take all camp at the same time also speed boost from Burnout lets her navigate faster and provides really good ganking possibilities.
I will add jungling section as soon as possible.

Some screens