Tryndamere is one of the champions I see that loads of people are struggling with, and yet worse, in my opinion builds him incorrectly, stacking thoose awful, awful Avarice Blade.
By the time "Avarice blade stacking Tryndamere's" have reached their fourth Avarice Blade and Boots of Speed, I have got my first B. F. Sword a Zeal and Boots of Speed.
And when they get their first B. F. Sword i'll nearly be done with my Infinity Edge!!
And I will also be significantly faster than they are.
And of course all this assuming we get the same amount of money.

So you'll probably see that my guide differs from most, since I choose not to use Avarice blades. Or start with any lifesteal items.

Also, this is my first guide, so constructive critics is appreciated.

I myself use a full crit page in my runebook, since this adds the little extra early game, that mostly will get you feeded early on, giving your team a huge advantage.

3x Greater Quintessence of Malice
9x Greater Glyph of Malice
9x Greater Mark of Malice
9x Greater Seal of Malice
Which adds up to ~30% at start without items and masteries.

Alternative Rune page.
3x Greater Quintessence of Malice
9x Greater Glyph of Malice
9x Greater Mark of Desolation
9x Greater Seal of Malice
This leads to ~22% crit at start and 18 armor pen without items and masteries

Offense:
3/3 Deadliness - Need I say much? 2% more crit.
1/1 Cripple - Longer duration exhaust, and decreases targets armor and magic resistance by 10.
3/4 Sorcery - Reduced cooldowns, always good.
4/4 Alacrity - Aspd a good boost for early game.
3/3 Sunder - 6 Armor Piercing, especially useful in early game.
3/3 Brute Force - 3 More damage, also useful in early game.
3/3 Lethality - 10% more crit damage, this mastery was made for Tryndamere.
1/1 Havoc - 5% more damage, yay.

Utility:
1/1 Spatial accuracy - Teleports faster, and less cd.
3/3 Good hands - 10% less dead, well not that useful
4/4 Awareness - 5% more xp
1/1 Greed - 0.1 more gold per sec, yay.

Which now adds up to ~32% crit and 6 armor penetration at start (~24% with Alternative Rune page and ~24 Armor penetration) without items this is.

CORE BUILD
Start with Brawler's Gloves (With this at start you'll have ~40% crit, ~32% with alternative Rune page.) and upgrade to Zeal (2% more crit, but adds attack speed and movement speed)

Then continiue with Boots of Speed

After this buy an B. F. Sword if you can afford it which becomes Infinity Edge
or you buy both Pickaxe and Cloak of Agility, and always start with Cloak of Agility when you choose between thoose 2.

Then upgrade Boots of Speed to one of the following

Berserker's Greaves - If the enemy team has got no hard CC

Mercury's Treads - If the enemy team is mainly casters, or if they got hard CC's, examples on when to get this is: against Ryze, Morgana, Amumu, Sona, Sion, Veigar, Annie, Anivia (And ofcourse many more)
NOTE: Due to large amounts of CC in the game, this will most likely be your boot of choice every single game.

Then upgrade your Zeal to Phantom Dancer
By either upgrading it the whole step directly or buying the needed items in this order
Zeal -> Cloak of Agility -> Dagger -> Phantom Dancer


SITUATIONAL STARTING ITEM

Youmuu's Ghostblade - Okay, I have been telling you of never getting this item, well, somewhat wrong of me, even though I do not recommend you start with this item. This item has proven it's worth as a backup item, I got this a game where I had no choice than to build it, my early game was incredibly bad, and by getting this item, I got somewhat useful before my IE, though, if your early game is good, do not get this item.

And when building this item do it in this order.

Brawler's Gloves -> Avarice Blade
After you have built your avarice blade (ewh, I really don't like this item...)
just buy a simple pair of Boots of Speed
After this buy a Long Sword and upgrade to The Brutalizer
After this finish you Youmuu's.

After you're done with your Youmuu's buy a Zeal because youmuu's alone isn't that much crit chance and we love the extra Movementspeed that zeal offers us.
With this done start building towards your Infinity Edge
And from here one you can decide wether to finish your Zeal into a Phantom Dancer or just go on with anything else.

TL;DR: get this item first when you have a really awful early game, otherwise don't.


Damage Items

Last Whisper - Just an awesome item to negate some armor, 40% to be exact, and also adds 40 damage, not a to bad exchange from the former 10 attack damage 40% aspd and the unique to 40dmg and the unique.
ALTERNATIVE ARMOR NEGATING ITEM: The Black Cleaver - Just as useful as last whisper if not even more against non tanks, and now is 55AD and 30%ASPD.

Sword of the Occult - If you just never stop killing the enemies.

The Bloodthirster - If you feel like you need lifesteal, but should not be needed, nice though if you are in alot of 1vs1 fights with melee champions, and adds up to 100dmg and 25% lifesteal after 40 mobkills. This you should get as a late game item though!
ALTERNATIVE LIFESTEAL ITEM: Stark's Fervor This item is extremely good if you have 1 or more strong physical carry! You get an lifesteal aura, an attack speed aura and you remove some of the enemies armor! Extremely nice! You should probably get this if no one else is going to.


Tankish Items:

Force of Nature - Gives hp regen, GREAT magic resist and movement speed.

Banshee's Veil - Despite the fact it gives mana (dead stat for tryndamere) It does give HP, MR and an epic spellshield that can save the day.

Spirit Visage - Great and cheap item for some extra MR and HP, also the passive is good for your bloodlust, not superb, but good.

Heart of Gold - A great item early on that gives a little survivability and Gold per 5, that can be built into the next item.

Randuin's Omen - Good active that makes all enemies near you while you smash them to pieces and still surviving due to the added hp and armor.

Frozen Mallet - If you still have problem with enemies running from you, also adds a nice amount of hp, and a little damage.

Thornmail - Useful if enemy team has got to much P-DPS, remember, you are not the tank!
Force of Nature - Useful if enemy team has got to much magic damage, remember, you're still not the tank!

Guardian Angel - Combined with your Undying Rage, and maybe a zilean, enemy team is going to cry.

Atma's Impaler - If you get any hp items, this might be particularly useful, and adds crit, and some armor for more survivability.

Warmog's Armor - really nice if you feel you lack hp. But should always be combined with Atma's Impaler

Bloodlust This skill is as said, your bread and butter, it's not kidding your best skill.
Many will argue with me over this and claim that undying rage is the best skill, well sure, undying rage will save your life alot.BUT, bloodlust will heal you for an impressive 400 when maxxed at lvl 9 (champion lvl 9 not skill lvl 9) and is not to toy with on a hard carry, as it does not only heal him for an impressive amount, it does aswell boost his damage by 24 and his critical damage by 32% and this is a tremendous boost to your damage and also keeps you in your lane for so much longer, especially since you'll have 30% crit at lvl 1 and easily 50% crit during lane phase without being in danger of dying.

Mocking Shout This skill make you so much better in 1 versus 1 situations, ganks, teambattles, detecting invis units and chasing and why this is so has easy answers. First of all, this skill slows anyone with their back turned to you (pretty much ever situation where you chase someone or gank someone as they will be running) and it decreases their current damage with 100, and this fits pretty much all the scenarios written above.

1vs1: You decrease you're opponents damage by 100, making you have the upper hand if it's against a p-dps character and if it isn't then mages will drop their spells and run, since you wont die because of undying rage, and when they run Mocking Shout and they can't run for to long with an furios tryndamere with undying rage running (+50% crit chance) smacking their *** with a badass sword.

Ganks: you remove their ability to fight back if they are an p-dps champion and you slow them if they don't fight back, how simple is this?

teambattles: You remove a total of 500 damage from their team! and surely making some of their champions go negative damage! So much for their carrys, you just removed 2 of their B.F's (100dmg) for a few seconds, well wtf, they'll hit like flies, this truly can (and will if used smart) change the outcome of every teamfight you participate in!

Chasing: You slow them and since you're probably faster than them without the slow, and if they flash over a wall, you follow with Spinning Slash.Detecting invisible units: That evelynn or that twitch go missing on the map? And you figure you would just clear this minion wave after the teamfight with low hp and Undying Rage on cd, and you notice that Mocking Shout is up and ready to cast, cast it and get the **** out of there, run to a wall use Spinning Slash to get over it and keep runnning!

Spinning Slash This skills scale poorly with damage but good with ability power, but doesn't matter, since you won't be getting any. As this skill doesn't scale good with damage, you shouldn't use it for farming, the thing you should use it for, is to travel! So, being ganked? Spinning Slash over a wall. Chasing someone? [spell=Spinning blade] and Mocking Shout. Seeing a opportunity to score a kill under the tower? Spinning Slash in and use Mocking Shout, hit once or twice, or how many times you need to score the kill, and then because Spinning Slash get 2 sec less cooldown every crit, you can now use Spinning Slash to get out again. So as you may have noticed, use Spinning Slash only to travel! (except the occasional clearing of low hp minions).

Undying Rage Makes you ignore death for 5 seconds, and gives you bloodlust stacks, the thing you plan to do during theese seconds is the most important decision you'll make as playing tryndamere, theese few seconds will come every now and then, and the things you do during this time is what
separates you from the noob tryndamere's and the good ones. I will give you a few hints on what to do now, if there is 3 or less enemies that has less hp than you do, then kill them! or atleast kill 2 and get away, except one of them is teemo, well sure you kill him in a hit or 2, but his blinding dart will
make you miss 3.5 seconds of your ulti! And his posion will be a major cause if you die when undying rage is over, and if there's a teemo, the chance of there's mushrooms is almost 100% and stepping on one of theese when you get away with 100~ish hp is so annoying! other than that, you don't want to engage if
there's 2 or more hard CC's or if there is some sort of DOT on their champions or if there's a tank among them. And remember, i'm not saying stay out of teamfights because of this, but rather than that saying that you shall not try to solo them if there's one of theese.

or summoner abilities, there isn't alot of choices, theese are the only ones you should pick.

Exhaust - Good since it slows them, and makes you hit harder, also changed so mages do less damage with it, and does not completely remove phys damage.
Teleport - Good for when in need for a fast return to your lane, needing to switch lanes because your teammates died or just to push more effective.
Cleanse- You will get stunned, mostly when in Undying Rage, so cleanse to be able to escape.
Ignite - You will have runners, and many close calls, and this might get a kill that you just wouldn't without it, but I recommend Exhaust over it.

But, you shouldn't pick:
Exhaust with Ignite
or Teleport with Cleanse

NOTE: Theese calculations doesn't make any sense at all, just a way to express my opinions.
If i put it like this, the offensive spells ( Exhaust Ignite)are two positive numbers.
And the defensive spells ( Teleport Cleanse) are two negative numbers.
You're aiming to get as well balanced as you can get get, and 0 is the balance.
so if two offensive (+1) is used then you get 1+1= 2
but if you use two defensive spells you get (-1)+(-1)= -2
and if you mix either, you get (-1)+1= 0

Yeah, you could still do extremely well with 2 defensive or 2 offensive, but I recommend you use one defensive and one offensive.

+ High damage.
+ Great farmer.
+ Slow and damage debuff. ( Mocking Shout)
+ Great escaping/ initiating. ( Spinning Slash)
+ Easy firstbloods

- Will get extremely focused (since you will be the best Physical DPS in your team 9 times out of 10).
- Going down fast when stunned or silenced.
- Hard to master.

You could start creep jungling as soon as lvl 6 if you got Zeal and Berserker's Greaves, start with killing the lizard, and make sure you heal with Bloodlust whenever you need to, but only heal if you got 8 stacks, or if it's really necessary. When you got the lizard buff, go grab the golem buff.
"WAIT WHAT? IT'S MANA REGEN FOR ****S SAKE."
Yeah sure, it's mana regen, but it's also cooldown reduction, and quite heavy reduct aswell, making your skills more spammable = Heal more, faster travelling, more slows/ dmg debuffs and more frequent use of your Undying Rage. And with theese buffs you can go ahead and kill the dragon for the 130 extra gold for your team, if you need help, just call for help from mid or bot lane.

Farming is really not a trouble with Tryndamere if you just get in range of the creeps, just time it well, and you will keep your Bloodlust up and get gold, never ignore farming, if you do that, you are about to fail. And as soon you get your B. F. Sword, you don't need to worry about last hits, since every hit is a last hit (Not really, but very close to anyways, and you don't need to time the hits very much)

The laning partners you really want are as following.

Janna - Knockup, slow, heal and attack boost/ Shield. Really awesome lane partner.
Blitzcrank - Grab, knockup.
Every single stunner, slower, fearer - Disabling them, DUH.
Zilean - Haste/ Slow, and revive.

Tryndamere is (IMO) the funniest melee character, with a somewhat high survivability 1vs1, will outdamage any physical dpser and can make attackspeed stacking heroes useless! ( Mocking Shout)
With this guide I hope you can improve your gameplay with Tryndamere, reaching a high KDR ratio, and change the game for your fellow mates.

And btw, thanks for reading the whole guide, and since you already have taken the time to read it, you might aswell comment and vote ;)