Shyvana is played as a tanky dps. This generally means that she will be quite tanky but will need to build damage items as well to be a threat late game. Tanky dps characters are also called bruisers and will usually follow the idea of being able to soak a lot of enemy damage while being able to dish out quite a bit as well. Their main tactic during teamfighting is to jump on the enemy carry [AD or AP] and zone them out from the rest of the fight. There are several intricacies involved in this and teamfighting isn't as clear cut as this intro will suggest. I'll explain teamfighting with Shyvana in greater detail in later sections, but make note that unlike a true tank, Shyvana does not offer any hard CC, nor any inherent soft CC. Frozen Mallet becomes a must in order to provide even soft cc. Still, she is an excellent choice due to her ridiculous damage output and ability to torment enemy junglers, and in effect, the entire enemy team. Shyvana can be played as a top laner or jungler, and it is entirely up to player preference as she excels at either role, but each has its own responsibilities and methodology. This guide will focus on the jungler role.

PROS:

Extremely high damage persisting throughout the game
Dragon's Descent provides a long range dash that deals a significant amount of burst damage [good for excapes as well as engagements]
A lot of abilities deal AOE damage, especially in Dragon form
Ridiculously fast jungle time due to Burnout and Twin Bite
Supreme counterjungler

CONS:

No CC
Seemingly weak ganking ability
Can be kited hard
Reliant on teams to pick complementing champions to dominate early game

I take 21-9-0 because Shyvana get a lot out of the offensive tree. The bonus % damage and increased physical damage augments greatly and helps her already great clear time and helps facilitate her damage potential for late game. The rest of the allocations in the offensive tree are pretty self-explanatory such as the summoner augment to exhaust, 3% lifesteal, 6 armor pen, etc.

The 9 points in the defensive tree include the 2 points in tough skin, reducing incoming monster/minion damage, 3 points in bonus armor, 3 points in bonus health per level and the 1 point in bladed armor.

Marks:

I take flat AD marks because of the increased damage per auto attack as well as bonus damage on Burnout and good scaling on Twin Bite. During early ganks, the bonus AD will destroy opponents' health bars.
Armor Penetration marks can work well if you are against a heavy tank team, since you generally will not purchase any items that offer bonus armor pen.
Attack speed marks will also be a good alternative. They really help you get through the jungle, since more auto attacks means more Twin Bite.

Seals:

I think that flat Armor seals work best to minimize the damage taken in the jungle as well as during early game fights.
Attack Speed seals could also be used to give an even greater clear time, but I don't know the cost by health benefits versus lowering minion damage.

Glyphs:

Pretty much MR per level glyphs as they provide the best stats for a tanky dps jungler such as Shyvana.

Quintessences:

I use Vampirism runes to add 6% lifesteal to total 9% lifesteal without any lifsteal items. It helps a lot in keeping your health high while jungling early and eliminates the neeed to start with a Cloth Armor and 5 health pots for jungling. Increased lifesteal is also very helpful in fights.
Flat AD runes will help with faster clear times and do even more damage and can be used instead of the lifsteal runes.
Movement speed runes are also an option since the 4.5% increased MS will greatly help you to chase down enemies during ganks or run away faster as well. A solid choice considering Shyvana's lack of CC.
Attack Speed Quints are viable for the same reason that the marks are viable. Really fast clear speeds.

I take Exhaust with its mastery to win early game fights while counterjungling. It also provides a soft cc when ganking really early. Combined with the Boots, 3 health pots start, laners without boots will not be able to escape early ganks, even with flash, with the exception of any inherent dash/blink abilities that give a second escape or hard cc disables in conjunction with flash.

Exhaust is also extremely useful during late game teamfights. Since your job is to sit on top of the enemy carries, while your own carries rip apart their tanks and initiators, exhausting the enemy AD will ensure that you win in the trade. Exhaust is just generally one of the strongest late game summoner skills and should be taken on junglers that can get away with not taking flash, as it will guarantee that one enemy damage dealer will be severely crippled for several seconds.

Max Burnout first since it's your primary jungle clearing and ganking ability. Max Twin Bite second since it deals a very high amount of damage and scales very well into late game. Ranking it up also decreases the cooldown significantly; combined with the attack speed from Wit's End, this skill will be refreshed constantly during fights. Save Flame Breath for last since the other 2 abilities outclass it early [the 15% armor shred is most effective during late game]. Take points in Dragon's Descent at Lvls. 6, 11, and 16.

Start with Boots and 3 Health Potion. With Lifesteal runes, you will not need cloth armor to start with and will be able to gank after red buff without going B since your health will be above the 3/4 mark. Shyvana's ganks at early levels are really good, especially with the speed at which she initiates on opponents with Boots and Burnout active.

Then I take Wriggle's Lantern and Heart of Gold because Wriggle's helps you get even faster clear speeds. Heart of Gold is an extremely cost effective item on Shyvana since passive defensive stat bonuses, i.e, Dragon's Descent's passive, are hugely helped by flat health gains. Also, the gp10 will help generate you enough gold going into late game for bigger item purchases, and doesn't make you entirely dependent on successful ganks to acquire gold. Plus, it'll eventually be upgraded to Randuin's Omen, making it a nice early purchase.

The choice between Ninja Tabi and Mercury's Treads is entirely dependent on the enemy team composition, taking into account whether they have more physical or magical damage, the number of auto attackers, and the overall CC that they possess. I generally would not purchase Berserker's Greaves as the other boots offer way more for your role in the team.

Wit's End provides so many benefits to Shyvana considering that the cost is only 2k gold. In addition to the bonus dps provided by the base attack speed gains of the item, as well as the bonus 42 magic damage per auto attack, the increased attack speed helps proc her Twin Bite more often. Furthermore, the stacking MR gain on each attack is extremely beneficial in being able to survive during teamfights.

Frozen Mallet is a must on Shyvana. During the early game, you'll have red buff to apply the slow on enemies to chase them down, but when mid game hits, and your AD carry starts to need the red buff, you will NEED this item to stick to your targets. And since Burnout provides a ridiculous MS boost, along with your Exhaust , there is very little way to escape Shyvana.

Atma's Impaler is the epitome of the bruiser item. It scales off of total health and provides decent critical strike % as well as armor. It is extremely cost effective; get it and start killing everyone.

Your Heart of Gold can now be built into a Randuin's Omen, granting you even greater tankiness [by this time, your armor values well exceed 200], and give you an AOE active slow and debuff to enemy AD Carries. This item helps you sit on top of those carries and refuse to let them do any damage to your teammates. The active also allows Shyvana to initiate very well with Dragon's Descent and an AOE slow.

As a non-mana dependent jungler, you can literally start anywhere in the jungle and do whatever route you please. My preferred route is to have my team help me on wolves, and give me a huge leash on blue. The mid laner should never hit the blue buff more than 2 times over the wall as it will cause the golem to reset, losing valuable jungle time. I tell my side laner, either top or bot based on the side of the map, to give me lots of hits so that I don't have to smite. Typically, I will save my smite for the red buff. After the blue golem, you have a lot of options:

Can continue jungling your own jungle passively, completing wraiths, then red, etc.

Can be aggressively jungling your own jungle by taking red right after blue, which will give you level 3 after completing the camp, using smite on red.

Can invade the enemy jungle and take small camps

Can invade the enemy jungle and immediately take their red buff using smite

Can hunt the enemy jungler while lying wait in the bushes

Can take a look at the lanes to see if there are any unusually easy gank opportunities.