Master Yi is often called a "nooby" champion and hence all too often underestimated.
Master Yi is a highly mobile AD Carry who can also be strong in early game. His Highlander and Alpha Strike allow him to get in fights very quick, to chase fleeing enemies down and also to make himself scarce if it hits the fan.
In late game, if nothing's gone wrong, he's truly the enemy's horror, as you burst out damage like there is no tomorrow and even though you got little life, armour and magic resistance, through your lifesteal you're hard to kill.
The most effective way to play him is with the aid of a support on bottom lane, which grants you a lot of gold to get essential items, probably early kills -as your opponent adcs are usually very squishy at that point in time- and after all loads of fun!

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It's all arranged to maximise damage, that's why the main part of the points is spend in Offense: Fury to get attack speed right in the beginning and Sorcery for some cooldown reduction, which helps in the early game before you get Highlander to refresh. Deadliness simply gives you more attack damage (scaling), Brute Force as well, but flat and Havoc boosts your damage in general. Investing two points into Lethality increases the damage you deal when hitting critical and Frenzy your attack speed. Weapon Expertise (flat) and Sunder (scaling) amplify your armour penetration. Executioner finally lets you generally deal 5% more damage to targets with less than a half of their health.

In Utility we got: Summoner's Insight for the reduced cooldown of Flash, Wanderer for more movement speed and Artificer to get Youmuu's Ghostblade and possibly the Mercurial Scimitar, the Ravenous Hydra or the Blade of the Ruined King faster back into the game.

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• As a Carry your main task is to carry your team to victory in late game. Due to this fact we concentrate on scaling attack damage with the Greater Mark of Attack Damage and the Greater Seal of Attack Damage.
  
• To get some additional attack speed we use the Greater Glyph of Attack Speed.
  
• You can also change the Greater Quintessence of Life Steal shown in this build to Greater Quintessence of Attack Damage or Greater Quintessence of Attack Speed, but I think it comes in quite handy to already have some life steal in very early game.

• Flash ~recommended highly~ The escape per se, to get away fast, also over walls. But also when you chase a very fast enemy, like another Yi. It will definitely save your life more than one time! (If you aren't level 12 yet you can replace it by Ghost.)
• Exhaust ~recommended~ Very nice to chase or escape, especially in the early game period when you got neither adequate items nor Highlander.
• Ignite ~recommended~ Always good to give an opponent the last kick or just to scare him off.
• Cleanse ~recommended~ Very similar to the Mercurial Scimitar described in Items and can be combined with it or replace it.
  
  All other spells are not recommended for Master Yi, as they are more or less useless or ineffective to him.

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Due to the fact that this build is an AD build we maximise Wuju Style first, but I recommend to invest the first point in Alpha Strike as it deals 100 magic damage, which you won't get with Wuju Style; in addition you can risk an early tower dive with it since you can simply flash out after your kill (take care of champions with the ability to stun!).

Every time you can't skill Wuju Style (lvls 4+8) you skill Alpha Strike as it is a pretty useful skill: Not only it is the perfect skill to farm effectively and to dive under turrets, as mentioned, but also is (with a little bit of practise) the ideal way to dodge various skill shots such as Nidalees Javelin Toss, Luxes Final Spark or even Kog'Maws Icathian Surprise. It's also a nice skill to escape: Just aim an enemy unit on your way (e. g. under an allied turret) and you'll get faster into safety. If the enemy unit is an champion with low life you could also get another kill. This works over walls as well. Note that you'll get back to your initial point if the target is killed during your leap before you reach it.

Skill Highlander every time you may (as you almost always do with the ultimate ability) and Meditate if you have to.

You could also skill Meditate earlier in the game if you feel like needing it (a nice possibility - which won't work at higher levels - is to lure with low health enemies under turrets or to your allies, perhaps slow them and then simply Meditate :)

Spawn Items

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At first you should buy three Health Potions

and the Boots.

Core Items

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Finish the Boots to the Berserker's Greaves.If you don't know what to do with your money after you got the whole build, I recommend the Enchantment: Furor.

Then in order to build Youmuu's Ghostblade you buy Avarice Blade

and The Brutalizer (you can also get The Brutalizer later on if your lane and the ganks are without any armour, but don't forget the cooldown reduction).

Do not complete Youmuu's Ghostblade at this point, as you'll lose the extra gold you gain from your Avarice Blade.

Now you begin to build Bloodthirster, I recommend first to buy the Vampiric Scepter to be able to stay longer on lane and then the B. F. Sword.

Now we come to the critical strike part: You finish Youmuu's Ghostblade,

buy Zeal and finish it to the Phantom Dancer.

At last purchase the Infinity Edge.

If you don't know what to do with your money after you got the whole build (incl. situational item), I recommend the Enchantment: Furor and after that the Elixir of Fortitude and the Elixir of Brilliance.

Situational Items

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The most important situational item to know is the Mercurial Scimitar as it is your ticket to freedom and probably survival if the enemy team tend to stun you ( Highlander does only prevent you from slows) and even grants you a movement enhancement (which is the overkill combined with Highlander and Youmuu's Ghostblade ;). You'll need it if you try to play Master Yi ranked.

Another very useful item is Statikk Shiv as it grants additional attack and movement speed and completes your critical strike build from a moderate 70% to an almost absolute (beastly 90%) chance to crit without critical strike runes.

If you fight against enemies with high armour Black Cleaver could maybe fix it.

The Blade of the Ruined King is the item of your choice if your enemies still manage to get away or you need more life steal. You could also get one more time Bloodthirster, but you will probably not gain the full advantage of the item any more, in addition you're more useful in teamfights with the crowd control.

The last item I'd like to mention is the Ravenous Hydra. It also grants you life steal and attack damage, but the key points of this item are the unique active and passive which spread additional damage over all surrounding enemy units, which is not only useful to get pentakills with ease, but also helps in teamfights as it probably deals (much!) damage to the whole enemy team and therefore causes chaos in the hostile lines (just imagine you're a support and notice you get large amounts of damage, you will react to this); at best they will lose their discipline and run uncoordinated around/away.

Master Yis abilities are not difficult to use or land, but the timing is very important, since you are pretty helpless without them, if you activated them to early or wasted them for another marginal purpose. Always remember that you only get the cooldown refresh if you kill with Highlander activated, so especially in teamfights you must get the first kill. Don't play too defensive, but play restrained and don't let yourself be carried away to kamikaze attacks.
And, the most important thing... well, just have fun! :)

-CoC

Special Thanks

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Many thanks to jhoijhoi. Without her Build Guide I wouldn't have been able to do this.

Another essential guide for me was Astrolia's Column Guide, as I spend almost an hour searching for a solution to get all columns in one line before I read it.

Also helpful were the Ten Commandments of Writing a Guide by Laggermeister.

Thank you.