Tryndamere is your classic hack'n'slash character. He reminds me alot of the fury warrior in World of Warcraft, as he uses his own health to deal tonnes of damage (Highest in both games if you ask me :P) No poisons, no stuns, no magical shenanigans, only pure damage, right on the table.
You shouldn't try to solo or go without a tank between you and Tryndamere, as you will get to taste his blade.

If you want to brag to the cool kids in school, don't hesitate to go

Pros:

Insanely high damage late game
Works very well even with a small amount of Lifesteal
Doesn't do bad in the start if you play him right
Has probably the best ultimate in the game
Very good for farming

Cons

Very easy to kite
Hard to time ultimate right
Will often be the main target, as he's simply doing too much damage
His left arm is much weaker then his right arm
Too much testosterone for many to handle

I've chosen a very offensive rushing build for the obvious reason that Tryndamere is a very aggressive champion. I don't really see him as a utility or defense champion, as he can't heal others and he's a tank anyway with his ultimate >.<

In spite of most builds out there, I don't go for Critical strike chance only. I suggest you to fill up all your marks with your seals should be And so far I've done well with in glyphs and flat In Quints.

I haven't experienced a build suiting my playstyle better yet, but of course I'm only here to show you a possible path, if you find something else better, just chose to use that.

Simple, the lower health, the more you're raging on your enemies, by constantly critting on them, perfect for when you're in Undying rage.

Remember to build up 8 stacks before entering a fight. You'll cut them like butter and then heal yourself after you sliced 'em to pieces.

So they think they can run? Not from Tryndamere, catch up on them with , also usable in team fights, by reducing the enemy's attack power.

This is your flee/Chase/farmer abillity. Use it through walls, minion waves and champions.

The abillity that makes your enemies crumble, use it when you're on low health, if you're 1v1, finish your target, if your 1v5, use it to flee.
Perfect for finishing off others with 5 sec of complete invincibillity plus 4/6/8 stacks of bloodlust.

Wait for a mob of minions has gathered at one spot, then use your to most of them, this should get them down to half health, then finish them off with auto attacks, your crist will reduce cooldown on , if there's any minions left, use a through them.
You can start soloing minion waves already at level 8-12, if you're sure noone'll gank you.
Actually can solo Baron, if you have all the items and you are level 18.
Remember to have the red buff all the time.

What to take

The best spell for Tryndamere, as he's very easy to kite. He needs to get close up to throw out massive crits.

Best chasing spell, especially against tanks, as it with a point in also reduces the defensive stats.

Third choice, as it works perfectly combined with .

Good for escaping with ultimate and chasing for the final blow.

Not really sure where to put this, as it's only useful to finish off low health champions staying in lane for exp.


What to not take

Hmmm... I'll let you figure this one out yourself.

Give this to your support, you don't need to see if someone's in your path, before they are dead by your blade.

You've got your bloodlust and potions, you don't need heal to frigthen your enemies.

Your damage is high enough without - you need to give others a chance :)
Have a support use it, if it needs to be used at all.

Only for jungling. I don't jungle Tryndamere, so I don't need this.

You'll just let your enemy see that your coming, you need to come from behind as if you've just slaugthered their base and now coming to take them.

You could maybe take this against towerdivers, although not recommended.

Just not recommended, as it has long cooldown and you shouldn't really be able to die...