Tryndamere is easily one of the best carries in the game. His passive Battle Fury and ultimate Endless Rage allow him to perform as one of the best melee DPS in the game. Playing this champion well requires fast reaction time and a LOT of farming. Keep in mind using these items and runes make him very difficult to play early game, since he will not have as high damage output or tankiness as other builds might.
When laning Tryndamere, always take the top solo lane. Also make sure to always be attacking minons to keep your fury up. High fury allows you to farm creeps easier and harrass enemies easier. The lifesteal from the mastery and Doran's Blade should be healing you for about 20's early game when you crit, so try not to use your q Bloodlust that often, since it reduces your fury back to 0. When laning against a dps make sure if they try to engage you you activate your Mocking Shout before they get to attack you that much. Since you get stronger as the battle goes on, they won't Exhaust you untill you are low, when tht happens you can either run or pop Cleanse and continue to smash their face in with your huge sword. If they back off, dont chase them, keep farming minons. Tryndamere's top priority is to get as many minions as possible. Laning against a mage is usually easier, since Tryndamere's base tankiness is one of the highest in the game. Your Bloodlust heal and lifesteal allow you to outlast almost any mana-using mage, though energy or health cost champions are slightly more dificult to face, since they can endlessly harras you. If this be the case, try to not push that much and only farm to last hit. If you decide to play aggressive, remember to watch for ganks. If you want to Spinning Slash toward them, make sure you spin to the side of them if they have skillshots, or wait untill they use them before spinning near. Dont use Cleanse every time you get stunned, or snared, or even exhausted. Save it for a moment you might really need it, such as turret diving or getting ganked.
Start with a Doran's Blade since the extra attack damage from it and your q, along with the runes, can allow you to hit almost 250 crits level one. 6% of 250 is quite a bit of healing. A Vampiric Scepter isn't a bad item to start with if you want long laning capabilities, or Brawler's Gloves if you want more crits and an earlier Avarice Blade. Be sure to rush your Avarice Blade as soon as possible, since the gold generation helps you get your items much faster. Berserker's Greaves are needed early as well for the move speed and the extra attack speed. After those, Wriggle's Lantern can help you lane MUCH longer, with the lifesteal and bonus armor. Surviving ganks from junglers becomes easier as well. After wriggle's, get your Phantom Dancer, prioritizing crit chance items, not attack speed. Going for an Infinity Edge can be a good choice as well, but your crit damage runes pretty much already give you it's passive, and Wriggle's Lantern gives you enough attack damage. The attack speed and movement speed can help more. Of course, after the Phantom Dancer, THEN comes the Infinity Edge. By now you should have around 200 minion kills. You are now ready to enter teamfights. With an Infinity Edge ready, your crits should reach around 750. With only 5 items, 2 of them being core, you can see how well you'll start to scale. After the Infinity Edge, rush your Last Whisper as soon as possible. Once enemy tanks get a taste of your damage, they will most definatly purchase some armor items. Last Whisper makes these almost nonexistant. 900 crits against 300 armor can go to down to about 200's. Last Whisper will bring these back up to at least 500's again. Once you get your last whisper, tanks should no longer be that much of a problem, so you get your squishy destroyer- Youmuu's Ghostblade. Combined with it's active move speed, armor penatration and your current dps, you should be able to 3 hit enemy mages or carries, if not 2 hit them. After this, you can sell Wriggle's Lantern for a The Bloodthirster, then boots for another one if you like. With Youmuu's Ghostblade's move speed and Phantom Dancer's, you still get above 420 move with no boots, and your Mocking Shout makes it no problem to catch up to enemies. Of course, end your build by spamming elixers and oracle's elixer with excess money.