1. Your fully stacked fury gives you 40% crit chance at all levels! Make sure you keep it up at 100% by autoing anything, even wards.
2. The lower your hp the bigger your damage is. When you killed your opponent and want to take a tower, don't use Q or potions to heal, but instead keep your HP low to tear down towers and minions way faster. But watch out for enemy JG.
3. You can Q in mid auto to have the crit bonus on it.
4. When attacking or chasing enemies, always walk with them after auto attacking. That maximises your DPS.
5. Follow the ability max page and DON'T max R. Your R points give you practically nothing. The 10 cd is not worth it.
6. If you are getting poked out in the early game, feel free to spam your Q it costs nothing and heals you by around 60 HP.

In loading screen, you need to think if you can win early all in or not. If you can, try to cheese the enemy in a tri bush or top bush. Your lethal tempo has great strenght for all ins.
When you lane, try to do short trades by auto attacking and then E away. Do that to wittle down the enemy for all in.

When the first tower falls, you want to stay in lane and cs/take more towers/camps. As tryndamere, you don't offer much to your team in small skirmishes. You are much better to farm up for items and take towers for guarnateed gold and xp rather than waste time trying to aram.
With that said though, you will need to join the big fights for drake or just 5v5. You can hover over the area without actually fully pathing there and take camps/cs while waiting for a fight to break out.

Late game is pretty much the same as mid game. You want to group with your team when you get your full build or close to it. Splitpush near the main objective baron/drag.
Always jump onto squishy carries first! Even if you need to burn flash or/and ghost to get onto the backline, it is WORTH IT.