Tryndamere is an AD carry. He is able to turn the entire game himself.
His playstyle has changed alot since the latest patch, where Riot introduced Tryndamere to the new fury system.
I will try to explain the differences in his playstyle, now and then.

Please, read the entire guide before you rate it

I used to play Tryndamere with a crit chance and crit dmg runepage.
But after the latest changes to Tryndamere, he is able to get very high crit chance, without even buying crit items, due the new fury system. For every 1 point of fury, he gains 0.35% crit chance. So, you can get 100 fury, that gives you a total of 35% crit chance, 37% with masteries.
So it's limited how many items you'll need for him to crit all the time

Pros
+ Good carry
+ Fun to play
+ Good ultimate
+ If played correctly, he can be able to take 2v1, even 3v1 sometimes (depends on the enemies ofc.)
+ Able to overextend alot (easy to escape with flash and spinning slash)
Cons
- Relies on his team
- Squishy (Depends on the way you build him)
- Can be hard to time the ultimate perfectly
- Easily countered by CC's

As summoner spells I like to take flash and exhaust. Mostly because, if they have a trynd or another carry, you can easily take them down without loosing too much HP yourself.
You can take flash and cleanse if you want. But I highly recommend exhaust.

Flash is a must, you are able to juke most people with a flash+spinning slash combo

Due his change to fury, I like to play tryndamere with attackspeed seals and glyphs, so he is able to gain fury faster.

I take armor penetration marks and quints to destroy tanks in seconds, aswell as squishies get more squishy.

This might sound like you are betraying your team, but it helps you win the game most of the times.
In teamfights you should wait to go in, until they have wasted their CC's. You should jump in from behind, to take out their squishy carries, one by one.

I hope this build helped you to understand the new improved Tryndamere a bit more. And I hope you have a lot of fun games playing Tryndamere. Remember, this build doesen't show how to play him everytime, it's just how I mostly play him. My builds depends ALOT on what the enemy team look like.