Welcome

This build is personally built by me, I have been studying Tryndamere for a while now and discovered different situations and fingured out how to handle them, and now I want to share them.
This build is greatly thought through, and I will appreciate criticism if you think there's something wrong with it.

The runes is a rather simple mechanism, as you know... Tryndamere's passive ability Battle Fury gives him great critical strike rating.
Therefore I have put the runes into that ability, so that he can cut through the enemies armor with some critical damage, and so he can take some damage too.

The armor penetration is the cause to your high damage towards enemy champions, with your damage items too.

Basically, it is situationally what my 5th item is. Usually, it will be Last Whisper, because usually, you meet some tank nor AD carries, which don't happen to take armor pieces.
So, during a late game where you have got your items, you will be able to cut through your enemies with only a few hits(If your Fury is full), plus with this built, you will not only be good with your team, but you might as well get tripple,quadra or penta kill, because of the double Bloodthirster. If it happen that you get quite low on health, you can use your Bloodlust to get some back, and if Undying Rage ain't on cooldown, you'll get the lost Fury back.

The Masteries talks for themselves. You get a lot of damage output and some armor and magic resist.

The summoner spells varies on what you prefer to play with, but I take these for two reasons:

Flash is not only good for you to flee, it also helps you chasing another champion. If it happens that you want a kill and the enemy is rather low, you can use Flash followed by a Spinning Slash, that way you can get to your enemy and get a kill on him, if he's running too fast, you can use your Mocking Shout to slow him, and if that ain't enough, you have Exhaust.

The other reason with Flash is that you can flee as said above, and Exhaust can be used for that just as well, if you're being chased you can pop it on the enemy champion, they won't most likely get you, as your Spinning Slash can get you through walls, or just straight forward.

So, in total. This build allows you to deal a high amount of damage and it also allows to solo a lot, including Baron.

Please comment with your responses, questions, what you like and what you don't like, and I'll edit it if there is any mistakes and updates.

Good luck and have fun playing Tryndamere!