Hi and welcome to my build!
My name is Psychrom, and I'm a Silver V player.

Before you rage on me, please try the build at least once. It's not for everyone, nor will it ever be.
Thanks.

The runes are a funny story. I used to main Master Yi back when I was first introduced to League of Legends. I never really knew how to play, and I eventually earned enough IP to buy full attack speed runes. -.- Dumb, I know. I eventually moved away from Master Yi and more towards Ability Power champions like Evelynn, Teemo, Ziggs, etc. The page hasn't had much use, but it serves it's purpose now.
The additional attack speed helps her take down the buffs early game, as all of her damage will be on hit. The 7 additional damage does help as well. Just don't rely too hard on these runes as they don't help much till late game when the additional attack speed could mean an additional attack just when you need it most.

The masteries have been built to fit her needs. You may realize she is a bit mana hungry, and that the build does really have much Mana Regeneration. The 3 points in the utility tree help a little bit there. The offense tree is where the other 27 points go, for good reason. An AD champ needs AD help, right?
You may ask, "If Irelia is jungling, why doesn't she have any points in the defense section, or at least two points in Butcher?"
The simple answer is: She doesn't need it.
She deals an additional 15 true damage from level 1 with her Hiten Style. Those points are better off giving her additional damage elsewhere. Furthermore, Irelia doesn't really need much defense as she will out DPS any AD champion, and burst down any AP champion. Target the strengths to outshine her weaknesses.

Here is where a lot of you may have problems with me.

We start a normal jungle start. Hunter's Machete, and five potions. Get a leash on red, it's the best way to start. You should only use a single potion for red, and then move to Wraiths. Use another when you're done, then head to Wolves. Another potion after wolves, then wait for Smite to be ready, or nearly ready (on cool-down around 5-10 seconds) before jumping into blue. Use those potions BEFORE you enter blue that way you don't accidentally die. After taking blue, you should head back to wraiths before heading back to base to get a spirit stone.

If you feel that you need more Attack damage late game, or that you're extremely Mana hungry, it's okay to buy a Tear of the Goddess. This comes in handy late game, though I rarely end up purchasing it.

ALWAYS, ALWAYS, ALWAYS buy potions or wards with extra gold you have left over.

Gank whenever needed or possible. Rush the Spirit of the Elder Lizard for the extra true damage. After you have this item, jungling becomes a breeze and buffs fall to your blades like snow in the winter.

I suggest the Boots of Swiftness because ganks are so much easier to achieve. Nothing like a fast Irelia sprinting into lane with her Q ready to initiate, and her E to slow. Follow up with her W to deal crazy amounts of true damage mid game. 100 true damage is 100 pure damage, whereas 100 attack damage is about 60 pure damage. Don't underestimate your power, but don't overestimate. If you die often, it's because you're overestimating yourself.

Nashor's Tooth as your third item adds roughly 23 magic damage per attack, plus it's base stats to your arsenal. Not a bad choice for you, damage wise.

I choose Hextech Gunblade next because it's just perfect on Irelia. More Ability Power means more on-hit damage with Nashors, as well as a slight increase in damage for her E and R abilities. It adds damage and lifesteal as well, which isn't a bad thing either. Furthermore, it adds 20% Spellvamp, and an activated slow.

(Here I'd like to add that Spellvamp affects ALL SPELLS, meaning any activated spell that does damage, not just spells that deal magic damage. All of your spells, except Hiten Style, will heal you for a partial amount.)

Wit's End should be self explanatory at this point, adding more on-hit damage, a bit of Magic Resistance, and the Attack Speed.

Once you have the previous 5 items, your 6th depends on how the game is going. If you're doing really well, go ahead and get a Phantom Dancer. If you die easily, then you may want a Spirit's Visage. At this point, your ganks should have your team well off, and it's usually not needed.

I don't recommend anything pertaining to attack damage, such as a Bloodthirster, or an Infinity Edge because it's honestly not needed.

"But what if I'm fighting a Teemo? He can just Blind me."
The blind doesn't last very long, and your Q, E and R deal enough damage to take any Teemo out before he can kill you. Let him hit first, then stun him with your E. (He has to have a higher healthpoint percent than you to stun him) The blind will wear off and then you can take him out easily by auto-attacking, then Q-ing him as he flees.
"But what about a (insert tank here)? Won't they kill me because I'm kinda squishy?"
No tank is useful against true damage. It's okay to sideline-focus a tank. (Meaning, every now and then, attack him once or twice to get the true damage burn on him.)

The whole skill sequence is entirely up to you. I HIGHLY recommend starting with your W because of the healing it gives you. You pretty much start out with a Doran's Blade if you choose your W. I suggest the Q as a second point pick as it allows a damaging initiation for blue and a level 3-4 gank.

After that, it's bananas from there.

Flash isn't a must on Irelia, thought it is quite handy.
Smite is obvious.
Ignite is alright if you feel like being overly agressive, though I prefer flash or teleport when I jungle.
I have found Teleport to be pretty decent on her, assuming you have a team that wards often and well. Nothing is more satisfying that teleporting to a ward to secure a fast kill or three.

I built this Irelia build as an experiment then took it to real games and it does work. I find it very viable, just try it for yourself. 3 or 4 games and you'd be surprised how easy it is to jungle her.

Always let your team know where you're going, when you're going to gank, and how aggressive you will be. It sucks to miss a kill, or die trying to kill someone while your teammate is backing.

TL;DR Gank often, gank hard.

Ranked play is nice, because if you choose Irelia, they immediately assume you are going top lane. This allows your top lane to counter theirs, making your laners stronger in the long run.

Play safer, of course.

Remember that your Q refreshes if it kills ANY target. Q a minion and then Q your actual target for easy, long distance kills. Or Q like crazy for a fast Pentakill clean up.

Your E only stuns if your opponent has a HIGHER health percent than you do. Sometimes allowing them to hit you first gives you the ability to deal more damage, as they're stunned and taking damage and you're not. The slow is sometimes enough, however. Master this concept.

Always activate Hiten Style when you enter combat. Your True damage stacks with your Spirit of the Lizard Elder.

Use your ultimate when enemies start to run, or if you notice your health is low. It heals for a decent amount. It's got a medium cooldown, but don't be afraid to use it when you want to.