Armor pen is needed with Tryndamere he doesn't need much extra attack damage because he can crit really well even without runes give him extra.
Attack speed is needed Tryndamere to get as much damage as possible.
Armor to reduce damage from minions and harass to stay alive.
Attack Damage really increase amount of damage he can do.

You need to get Bloodlust up as fast as possible it gives you damage, restores your health making it a lot easier to harass and survive in lane. Should be leveled by level 9.

Spinning Slash is a strong move that allows you to get close to the enemy and you can lower the cd by getting crit strikes. Should be Leveled up by level 13.

Mouthing Shout cuts the attack damage of the enemy and when they are not facing you it slows them a very great amount should be gotten early but leveled to full last.

Undying Rage is needed with trynd it makes him so he cant die and if used at the right time can make the difference between dying or killing the enemy.

Start with Brawler's Gloves and 2 Health Pots for the early Avarice Blade and extra health if needed in lane.

Get Berserker's Greaves for movement speed and attack speed so you can chase people down and kill them fast.

Get Zeal and Vamp Scepter for extra movement speed, attack speed, crit chance, and life steal so Trynd can farm and gain his life back faster to make it harder to be harassed by the enemy team

Infinity Edge NEEDED for Trynd he is basically a right click you die it increases his damage by a lot and his crit is already high but he gets more from this and the damage he deals with the crit gets upped as well by 50% making it really easy chop through peoples health bars.

Phantom Dancer is a must for Trynd if you don't have this you are playing trynd wrong. gives much needed movement speed, a lot of crit strike and attack speed. When you have one phantom dancer, with an Infinity Edge you can reach 100% crit at full rage. Which is really
hard to counter. I recommend two of them because it allows you to attack extremely fast and get your crit chance to 100% faster and gives you more movement speed to make it near impossible for enemy champions to get away.

Bloothirster Life steal is needed with him, otherwise you get destroyed by the other team if they have a thornmail if you dont have it you basically kill yourself faster than you can say ****. Which is why I recommend two of them.

You can go Warmogs and Atma's Impaler which makes you more tanky but I have never been able to use that well to be honest and I have never seen a strong tryndamere that had that build.

YOU ARE A CARRY
you need a lot of farm, the Avarice blade helps with gold a little but you still need a lot of farm that is true for every carry and shouldn't need to be explained.

Pros

Very strong early game
Can self heal
Has a high Crit chance when rage is up
His ult
Great harass
Great at escaping

Cons

Squishy in the beginning
Hard to farm when fighting ranged
CC destroys Trynd

I am new to making builds so any suggestions to making this better would be great I will do my best to keep this up to date.