This is my Tryndamere, with my build i've been know to melt tanks, other Tryndameres, casters, and even ace teams single handedly in 3v3s.

With your runes you basically want to focus on Critical Strike Damage. You can get strike chance if you feel it's necessary, but i like to just worry about that when I'm in game with my Item purchases. I try to split my runes up between Crit Damage and Attack Speed.

In my Item purchase build you can see that there are more than items that you would have, well Some are situational. If you are facing 1 or more tanks, make sure to get a last whisper and a black cleaver after you get your Infinity's Edge. Late game you almost always want to sell your Avarice Blade, that way you can replace it with something with a good bit of attack damage or maybe a Phantom Dancer. If you need the few seconds of movement speed, keep your Avarice Blade and buy Youmuu's Ghostblade. But more than likely a Bloodthirster or Black Cleaver would come in handy a lot more.

Make sure to focus on your Bloodlust early game, Tryndamere is very easy to kill for ranged champions and champions with stuns during early games. So focus on getting Bloodlust as much as you can to keep your health from getting too low.

I normally use Ignite and Exhaust. For champs with very low health near their turrets, Ignite them and run away so you can get an easy kill and not worry about it. Exhaust i mainly focus on using the most early game during ganks and if i'm getting chased i use it on either the fastest or the closest so i can hopefully get away.

Usually, because of Tryndamere's ult, in team battles, enemy teams focus Tryndamere first to try and keep him from getting too many kills. You can either wait for your team to jump in first then go in and catch them off guard, or you can take advantage of it and try to take out as many champions as you can while your team does additional damage behind you.

Thank you for reading my build on Tryndamere. Use the item's I posted on here wisely, and remember their are more items for certain situations. I hope you use this build to find your own new way to use Tryndamere and crit some teams to death.
Good luck and have fun.