With his rage based remake, Tryndamere is a new animal. While you could certainly build the usual crit machine, it is now much more efficient and better to build attack speed. His rage bar that fills per hit increases his crit chance, while he gains attack damage from missing health.

For runes, I decided to go for armor pen and attack speed. Because I jungle Tryndamere, the attack speed allows me to fill up my rage bar faster, allowing me to heal more with Bloodlust, and have a higher chance to crit with a full rage bar.

21/0/9 masteries based on jungling.

My items are focused mainly on attack speed. I get Wriggles for the jungle and Berserker's greaves. Then I get Zeal. So far, a fairly standard build.

Next, I get Guinsoo's Rageblade. Generally an item picked up by Jax and other hybrid champs, I find that the Rageblade is excellent on this new Tryndamere. Giving him extra attack damage, Ability Power, and Attack Speed, Rageblade synergizes really well with Tryndamere.

Then I grab Stark's Fervor for more Attack Speed and lifesteal.

Grab Infinity Edge and then sell Wriggles for a Black Cleaver and rip through your enemies.

At full build, you should have 1.974 flat attack speed, add another 32% for full stacks of Rageblade, 296 attack damage and 51 armor pen.

Max out Bloodlust first, putting a point in both Mocking Shout and Spinning Slash. Grab Undying Rage at level6, and the finish out Spinning Slash and Mocking shout last.

I like to get Exhaust and Smite as my standard jungle spells, but Flash or Ignite would work in place of Exhaust.
Don't bother with heal or teleport, or anything else. You simply will never need them.

Pros: Lots of damage
High Attack Speed to fill out the Rage Bar quickly
A very fast Jungle
Cons: Weak against CC heavy teams
Need to be aware of counter junglers.

My jungle route starts at wolves and follows the same route as Gangplank and Nocturne, with taking out Lizard buff before Golem.
Jungling is good for Tryndamere as it allows him to farm without too much harrasment. Also, getting kills off of ganks will start to snowball him into an unstoppable force. Just spin out of the bush, hit Mocking Shout, and cut them down.

Remember that this new Tryndameres is more attack speed based than crit based. Also be aware of when Endless Rage is up. Don't charge in without it or support. Also build accordingly. If the enemy team is heavy on CC, grab Mercury Treads or Cloak And Dagger, and also grab a QSS.

Remember that this is a jungle guide, and that Tryndamere is not the standard jungler. If you are new to this part of the game, jungle with Warwick or Nunu or Nocturne to get used to the flow of it. You have to be much more precise in the jungle than in lane.