Hello, I'm DrunkAsianFather,
I'm Dutch so sorry for my bad english :3.
Personally I like to jungle in League of Legends, and I really like Jungle Tryndamere, so I'm going to show you how I play him.

Pros
Very strong fighter
Very quick jungler
Amazing ganker
Can spin trough walls
Extreme slow
Can heal
Fed Trynd = GG
UNDYING RAGE

Cons
Item dependent
Right-click champion -> stopped by Exhaust, Blinding Dart, Wither etc.
Building full damage so quite squishy
Early game farming is ****py -> that's why we are jungling!

Passive: Battle Fury
This is a very nice passive, it will boost your damage output massively since you get a lot of critical strikes with it.

First ability: Bloodlust
This will give you damage when you are low, but also heals you, so use it when you need HP, otherwise just regain health with lifesteal since you will heal a lot because your fury bar gives you crit chance. I take Bloodlust at level 1, and max it first for damage.

Second ability: Mocking Shout
Your slow, your CC, I don't use it a lot for the Attack Damage debuff but in large teamfights, use it! The slow will let you catch up, making Tryndamere a great chaser together with your third ability. I take Mocking Shout at level 4, and balance it together with Spinning Slash

Third ability: Spinning Slash
Your gap closer, your ''blink'', you can spin trough walls! Also it deals significant damage, and it's your only AoE ability. I take Spinning Slash at level 2, and balance it together with Mocking Shout.

Ultimate: Undying Rage
This is why you are Tryndamere, the ablility to NOT die. Use it smartly! You can use it in teamfights when you are near dead, try to lifesteal your health back or use Bloodlust to prevent death when escaping. I take Undying Rage at level 6, 11 and 16, whenever it is available.

Greater Mark of Attack Speed and Greater Glyph of Attack Speed
Attack speed -> this will speed up your clearing time in the jungle and builds up Battle Fury very quickly.

Greater Seal of Armor
Armor -> this will increase your survivability in the jungle, since it reduces incoming physical damage so early game jungle creeps deal less damage.

Greater Quintessence of Desolation
Armor Penetration -> You don't have a lot of Armor Pen in your other runes so take it for your quints.

A 21/9/0 jungling build, grabbing almost all the AD masteries in offense and armor and health in defense.

When jungling you should start with a Vampiric Scepter, Bloodlust and head into the jungle.
Ask your teammates to cover and check for enemy invasions, also tell your support to use Clairvoyance to check where the enemy jungler is starting, or if he is counter jungling you.Blue: checking invasions
Red: clairvoyance finding the enemy jungler

For your jungling path:
Take out the wolves with basic attacks (never use Smite on wolves.
Transition to the wraiths, use Smite on the big wraith and learn Spinning Slash, take out the small wraiths with your spin and basic attacks.
Then go to the small golems and auto-attack and spin untill they die, when reaching level 3 take another point in Bloodlust. Depending on your health, go to the red lizard and kill the small lizards and recall and buy Boots.

You typically don't want to use Bloodlust unless at the last part, since it will lower your damage output, and with a full fury bar and low health you will heal with lifesteal, so only activate Bloodlust at the small lizard from the red buff.

After that go to the blue golem and take it down with auto attacks, Spinning Slash and Smite. You might want to donate Blue to your mid laner (AP Carry) but Cooldown Reduction is always nice, and you get the experience. When reaching level 4 get a point in Mocking Shout. After that just clear the jungle. ALWAYS GANK if there is a possibility, you can't sit back and stay in your jungle all the time, your team needs you.

This is an old path, after the Volibear patch the jungle got nerfed, now your path should be: Wolves -> Blue -> Wraiths -> Wolves -> Red -> Small golems
Play Video

Tryndamere if very strong in his ganks, especially with the Lizard Buff. Always try to lead in with just basic attacks, if your enemies are running away catch up with Mocking Shout, if they blink away or are far away, use Spinning Slash to close the gap. Remember that you can easiliy dive your opponents, especially when you've reached level 6 and learned Undying Rage, eventually use Bloodlust to survive.

For summoner spells, I take:
Flash: Blink! Chase/Escape, sure you have Spinning Slash but combine it with Flash! It's a summoner spell I take for every champion.
Smite: You are a jungler, take Smite! Steal dragon, buffs, baron with it.

Alternatives for flash:
Ghost if you are a fan
Ignite to ensure kills, I personally never take ignite for trynd.
Exhaust CC = GG

Rally always good if you are trolling!
Clarity Hell Yeah!

CORE
Vampiric Scepter Mercury's Treads Zeal Infinity Edge

END GAME
Bloodthirster Mercury's Treads Phantom Dancer Infinity Edge Quicksilver Sash Black Cleaver

WHY?
Bloodthirster: Life Steal and damage, upgraded from your early bought Vampiric Scepter
Mercury's Treads: Movement speed and Tenactiy with a little bit of Mage Resist.
Phantom Dancer: Movement and Attack speed, Critical Chance. GG
Infinity Edge: Critical DAMAGE, and attack damage. A few basic attacks and you are dead bro.
Quicksilver Sash: Exhaust or any other annoying spell? Hell nah! Activate and nobody can stop you with Exhaust.
Black Cleaver: Armor becomes an issue? Not anymore.

OTHER VIABLE PICKS
Too Squishy? Warmog's Armor or Frozen Mallet
AD enemies? Thornmail
AP enemies with CC? Banshee's Veil
Tanky enemies? Madred's Bloodrazor

And this was basically it! I think I told everything I wanted to tell you about Tryndamere.
I hope you enjoyed reading my guide and I hope I'll see you on the Fields of Justice.