Tryndamere is a very powerful late game carry. His abilities are all very offensively based and can also be used defensively early game. He has great potential in chasing, running away and just murder. Murder isn't even the right word for it, it's more like R@pe. While he is so powerful, his early game tends to be more weak and easy to die. Before lvl 6, good players will always try to focus you down and kill you. However, if you follow this build and don't panic whenever you're at low hp, you should do fine in all stages in the game and carry like a boss late game.

For runes, you should take
9x Greater Mark of Furor which gives you critical damage;
9x Greater Glyph of Malice which gives you critical chance;
9x Greater Seal of Furor which gives you critical damage; and
depending on your personal playstyle you can either take
3x Greater Quintessences of Malice which gives critical chance;
3x Greater Quintessences of Alacrity which gives you attack speed; or
3x Greater Quintessences of Furor which gives you critical damage.

If you're just a starter in Tryndamere and don't want to spend all those IP into buying Tryndamere runes, you can always get plain Armor Penetration, Attack speed and Critical chance runes.

Since Tryndamere is so squishy early in the game, you want as much survivability you can get, so I get a Vampiric Scepter to start off, which gives me more hp back when i'm lower in hp since Tryndamere's passive gives him more critical chance. Then I go ahead and get Berserker Greaves and a Zeal. A lot of people will ask me why not rush an infinity edge? Because this is a late game build, and in early game you really want to depend on you vampiric scepter to help you survive. Then after building a Zeal, rush an infinity Edge. Once that IE is finished, you should be owning a majority of people if you play right. Then after finish that Zeal into a Phantom Dancer, then finish that Vampiric Scepter into a Bloodthirster. Once those 4 items are complete, they other team should probably surrender because you're owning too much for them to handle. If they don't, then just go ahead and get another bloodthirster, then another Phantom Dancer. For Tryndamere, if you get 2 Phantom Dancer, that will get your Critical Chance to about 97% with full hp. So you don't want to rush your 3rd Phantom so fast. Once it's so late in the game you have so much gold and you already have 2x Phantom, 2x Bloodthirster, Berserker Greaves,and an Infinity Edge, you should go ahead and sell your boots that's for noobs for another Phantom Dancer which will bring you to beyond Godlike.

R>Q>E>W

In the first 3 levels, get all three skills, then just level up your Q and R as a priority and max out W which is your Mocking shout last. REMEMBER mocking shout reduces damage as well, so don't be afraid to use it in a team fight even if they are facing towards you.

For summoner spells, Cleanse is a MUST for Tryndamere, simply because if he gets CC'ed he's useless and WILL feed. For the other summoner spell, either get exhaust or flash so you can perform a double flash with your Spinning slash or exhaust to lower your opponent's stats.

Pros INSANE late game carry
Everyone's scared of you
Very great farmer ( 1 hit creeps )
Very fun with great escape mechanics
Damage reduction
Average around 15 - 20 kills per game in normal games easily

Cons Horrible early game if not pro with Tryndamere
Very very hard to play when other team has CC or stealth
Tends to get focused early game - mid game

Tryndamere CAN jungle. Trydamere has decent jungling capacity and will eliminate early game harrassing in lanes. While jungling, start off with the golems, then go to the wolves, then smite the large warith then clear it. After head to the buffs, I usually go blue then red. By this stage you should learn your mocking shout already, head to gank other players and grab some easy kills.
Item builds are the same because vampiric will keep you alive, Tryndamere NEVER go back to base while jungling except for shop.