Hello, This is Martellato (EUW) and I am here with a guide, discussing about Tryndamere, a champion who has it all!
Tryndamere is a kind of champion who always make enemies whine, if played well.
His skills and passive match PERFECTLY and I mean it. It comes in a very low price for such AMAZING Champion [1350 IP (can you believe it?!)]. ] If you don't have this amazing champion, make sure you buy it ASAP or if you already have then we're ready to Rock (\m/_ OH YEAH!!!) Summoner's Rift!

Here's a quick list of terms and acronyms used throughout the guide:
AD = Attack damage
AP = Ability Power
AS = Attack Speed
Ar. Pen. = Armor Penetration
M Pen = Magic Penetration
LS = Life Steal
SP = Spell Vamp
CRC = Crit. Chance
CRD = Crit. Damage
HP5/MP5 = Health/Mana per 5 seconds
GP10 =Gold per 10 seconds
CDR= Cooldown reduction
Nuke = Champion who deals a large amount of damage within a very short period, then waits for cooldowns
DPS = Damage per second, Champion who deals damage over time through auto attacks
CC = Crowd Control (slows, taunts, fears, stuns, snares)

PROS:

+ Battle Fury +35% Critical Chance, now imagine with Infinity Edge your Critical attacks which is 100% of your auto attacks, deal 250% damage! Very high auto-attack damage

+ Bloodlust: Heal and a Maw of Malmortius, Isn't that JUST great? imagine having 0 life and still being able to fight thanks to Undying Rage

+ Mocking Shout awesome -AD debuff to surrounding enemies

+ Spinning Slash damage dealing dash :O!

+ Undying Rage This is hilarious! no matter how much damage the enemy is dealing to you (imagine 1v5) you can get at least a kill!

+Awesome skins! (Heck there's a Legendary one)

CONS:

-Very vulnerable to CC

-Takes a little time to be the OP one, I mean Tryndamere is Mid/Late Champ!

-Focused

IMMORTAL for god sakes)! Therefore, MR/MRS Summoner you can go O-F-F-E-N-S-I-V-E big time (yer be dealin about 3k dmg each second) so Sit tight and lets proceed with the guide... oh wait I have some more to add:

Summoner's Resolve - Extra gold on Smite?! who doesn't want that?
Alacrity - We need that like duh!
Weapon Expertise - 10% Ar. Pen.+ Runes + Sunder !
Lethality - Because Tryndamere Breaths Crit!
Vampirism - Some LS for start.
Veteran's Scars - +Health at start!

Greater Mark of Desolation

- My own favorite, awesome Ar. Pen., FAST jungle and easy kills on early ganks.

ALTERNATIVES:

Greater Mark of Attack Speed

- Tryndamere is an AD carry anyway, so more attack speed, which means more damage/sec

Greater Mark of Critical Chance

- More crit?! damn this is getting scary.. but is it? well i've got to say no, with the rest of the items such as Infinity Edge or Phantom Dancer you get a lot of CRC ( a lot i mean 100% with full stacks of his passive Battle Fury. and since >100% CRC wouldn't make any difference, I Wouldn't recommend!

Greater Mark of Attack Damage

- great starting item but well, wont be much of a help mid/late. Because i'm telling ya, Tryndamere is a late game boss!

Other marks are just meh~

Greater Seal of Armor

- I'd say its the best seal out there! just awesome, great against Solotop, Bot carry and jungle creeps! Haha

Other Seals are just meh~

Greater Glyph of Scaling Magic Resist

- Since AP casters gain Damage over time, It's not a a necessity to buy flat MR Runes. Yet you get enough MR to save yourself in team fights.

Other Glyphs are just meh~

Greater Quintessence of desolation

- My own favorite, awesome jungling, very high damage through out the game.

ALTERNATIVES:

Just like Marks, for Quintessences you may use:

Greater Quintessence of Attack Damage

- Good start + Bloodlust gives a very good auto-attack damage.

Greater Quintessence of Life Steal

- "why not zoidberg?" Lol! Well, This is a great help until you get your first Bloodthirster but with Tryndamere's heal on its Bloodlust, it won't be a necessity :)

Greater Quintessence of Critical Chance

- More Crit?! seriously!? You may want a powerful Tryndamere but you wouldn't be a sadist over it, would you? As I said this kind of Runes make Tryndamere's Crit chance go OVER 9000 (100% lol) and that'd be a waste of money~ xD

Other Quintessences are just meh~

Bloodlust >> Spinning Slash* >> Bloodlust >> Mocking Shout*

* Mocking Shout and Spinning Slash are necessary skills for ganking, yet Bloodlust is needed for more powerful jungling. So as to say after LV4 you are able to gank but tryndamere isn't the best ganker (compared to Shaco, Nocturne, Skarner, Trundle because of lack of solid CC, Mocking Shout is conditional, therefore you might need to dive turrets with the help of your Ultimate Undying Rage but since the Cooldown is pretty short, you will never have Ultimate cooldown problems, it's always up!

Here we discuss some other item choices

Defensive Items:

• Frozen Mallet: Frozen Mallet, is a great item if you desire HP, Frozen Mallet's passive which is slowing enemies on auto-attack hits, is really useful and is great in team fights. Yet I personally don't see the point of building HP items on Tryndamere thanks to the awesome ultimate Undying Rage.
  
• Warmog's Armor A lot of HP 1270 at full stacks and 40HP5 which can be replaced with one of Bloodthirster
  
• Trinity Force Trinity Force is a useful item on every AD champion (except for the mana it gives on Tryndamere) everything else, Specially the evolved unique passive from Sheen which is extra damage on each ability use + the Frozen Mallet's Passive, slows your opponents and after using Mocking Shout or Spinning Slash when they're running away, Gives an enormous boost of damage and secures your kill.
  
• Zeke's Herald Zeke's Herald
  
• Randuin's Omen
  
• Ionic Spark If you decide to purchase this item, it must be bought early game, that way 150 damage is good in jungle and against champion, but later when enemies build defensive and minions gain health, it goes haywire.
  
• Wit's End Against AP-heavy enemies
  
• Guardian Angel Very good late game item, decent Armor/Magic Resist and resurrects you upon death. If you decide to get some defense, this would be the best item for you!

Offensive Items

• Last Whisper more AD, more Armor penetration. these kind of items let you go through any armor, but you gotta watch out foe Thornmails on enemy team.
  
• The Brutalizer Just incase you need some Cooldown reduction and some ar. pen..
  
• Executioner's Calling Sometimes I build this in dominion. This item is kind of underrated and never seen people using it. Amount of Critical chance is low but the life steal is pretty decent for the gold you pay.
  
• Atma's Impaler Great item with Warmog's Armor or Frozen Mallet, specially the Critical chance is really useful.
  
• The Black Cleaver

Boots:

• Mercury's Treads The best Tenacity item in the game. Amazing against those with a lot of CC.
  
• Ninja Tabi For jungle and solotop against AD champions with low CC.
  
• Boots of Mobility For better ganks and faster jungling.

According to Lolking.net, These are the most common Spells on Tryndamere:

46.8% of summoners use Flash. Combined with Spinning Slash becomes a powerful escape pursuit mechanism

SPACE

40.9% of summoners use Ignite. A great way to secure kills in lane.

SPACE

40.3% of summoners use Smite Apparently 40.3% of people use Tryndamere to Jungle, But for Jungling, Smite is a necessity!

SPACE

28.2% of summoners use Exhaust Another great ganking item.

Jungle Route

Tryndamere is not Blue dependent but Red Helps him a lot. you may invade their red at first.

Highest Priority*
Second Priority
Helping Teammates

* Baron Nashor and Dragon must be warded after their initial spawn time which are 15:00 and 2:30

• Use your rage meter between camps, not during fighting. You’ll get higher heals and clear faster that way.
• When you use mocking shout, be sure that they have their backs turned.
• Like Xin, save your spinning slash until you have to use it. It has a shorter cooldown, but at the early levels you can just walk in, shout/attack and spinning slash when the ghost/flash away. This will help you secure kills more than just “sure wish I had spinning slash” will.
• Don’t forget you can use your ultimate any time no matter what. Stuns, silence, suppress, you name it. You can’t be killed!
• Practice what walls and terrain you can and can’t spinning slash through to make your escapes that much better .

• Vayne!!! Yes Vayne!!! With Condemn (stun) + Tumble and Final Hour when you're trying to escape
• Jax, Since Tryndamere is a champion that deals all of his damage from Auto-attacks, Jax can dodge with the help of his E ability Counter Strike and catch you after your ultimate is over with Leap Strike
• Teemo, with Blinding Dart, same reason as Jax

Also some items such as:

• Sunfire Aegis
• Frozen Heart
• Thornmail

• Tryndamere was designed by Guinsoo
• Tryndamere is voiced by Brian Sommer, Who also voices Nunu & Willump and Warwick.
• He previously held the title "The Dark Champion", but since the second official trailer, he now holds the title "The Barbarian King".
• In the August 31st 2010 edition of The Journal of Justice, it was written that Tryndamere was passed down his sword from his father, who had it passed down from his father before him, and so on. Tryndamere said his father's words of wisdom were "master using it and you can have this" - a quote from the original Legend of Zelda on the Nintendo Entertainment System, released in 1987.
• Tryndamere has strong similarities to Conan the Barbarian.
• Tryndamere's dance refers to Hopak, the traditional Cossack dance. His previous dance, before patch V1.0.0.122, was a reference from N'Sync's dance in their song, Bye Bye Bye.
• As of patch V1.0.0.122, Tryndamere now uses the Fury system instead of health for his abilities.
• Tryndamere is married to Ashe. However, Ashe says that they married for political purposes and not romantic.
• Sultan Tryndamere is a commemorative skin created to celebrate the launch of League of Legends in Turkey.

• Thanks to jhoihjoi with "Making a Guide" guide
• Also Hahano with "Who is your jungler and what does he do?" and "Jungling 202: Beyond The Basics"