Hi, I'm GeNeRaL, a Irelia main. I mostly made this guide because I noticed that pretty much all Irelia guides out there seem to recommend builds that I don't think are as strong as they could be, most notably by recommending Blade of the Ruined King as a core or semi-core item.

Irelia is a top lane fighter. She has very high base damages, a good dueling potential and good burst damage She tends to be chosen for her snowball potential and her ability to hard carry mid game fights. Our goal on her is to get our Trinity Force and start bursting as many enemy squishies as we can get our hands on. Irelia is one of the best champions for carry the mid game.

Pros:

Good mobility.
Safe in lane.
Few bad matchups.
Strong CC.
Burst damage.
Strong mid game.

Cons:

Mana Hungry.
Weak early.
Fall off in true late game.

A lot of people seem to think that Blade of the Ruined King is an awesome item on Irelia. These people tend to incorrectly believe that Irelia is an auto attack/dps champion in teamfights who is at her strongest in late game. She isn't either of those things. Irelia is strongest just after completing Trinity Force, and then slowly falls off. This fall off isn't dramatic- you're never useless, and you're often stronger than your lane opponent in late game. However, there will come a time when you're not able to dive the ADC by yourself and kill them.

Blade of the Ruined King doesn't help with that. That's because the vast majority of your damage comes from Trinity Force procs, Equilibrium Strike, Bladesurge, and Transcendent Blades. You don't typically get off enough autos in a teamfight for Hiten Style to have a huge impact. BotRK provides mostly attack speed, which you can't usually use all that well in a teamfight, and not that much AD, so it doesn't add to most of your reliable sources. Ravenous Hydra does help- it gives a lot of AD for Transcendent Blades and Bladesurge, and its active provides a bunch of additional burst. Youmuu's Ghostblade helps also- the armor pen amplifies your Trinity Force procs, which is extremely painful. BotRK is only good for splitpushing or DPSing down tanks. Ravenous Hydra does a better job of the first, and the second isn't your job.

Your passive is Ionian Fervor. It's quite a strong passive that gives you a little bit of tenacity any time an enemy is nearby, increasing to a really strong amount with 3 enemies nearby. This does stack with other sources of tenacity such as Mercury's Treads, although not perfectly.

Your Q is Bladesurge. This is a fast targeted gapcloser. It deals full auto attack damage, some bonus damage, and applies on-hit effects, so you can use it to deal an extra burst of damage with your Trinity Force procs. The part of this skill that makes it really strong for you is that is if last hits an enemy, it will reset its cooldown and return you some mana. It's really important, especially in the early lane phase, to always get the reset with your Q so you don't run out of mana and can always dodge enemy skillshots. You can use it to farm minions as well, just be aware that it costs you mana to do this. The refund is only a flat amount of mana, so as you skill it, it will refund for proportionally less. This skill makes you really really mobile once you max it, and as soon as you get Trinity Force you can 1-shot caster minions with it and Hiten Style, allowing you to jump all over the place as long as minions are there and Trinity Force isn't on cooldown.

Your W is Hiten Style. The passive portion gives you a small amount of health gain on-hit. When you activate it, it doubles the health gain you get on hit, and also gives you a large amount of on-hit true damage. This is where your DPS comes from and also where you get a little bit of sustain. You'll want to have this up any time you go in for an extended trade. There's generally no reason to use your Bladesurge on an enemy champion without Hiten Style active, unless your Bladesurge will last hit them without that damage. While the bonus true damage will not work on structures, the healing will- both the passive and the active portions.

Your E is Equilibrium Strike. This skill is incredibly important to use well, and knowing how to do that is a bit of an art. It deals a good bit of magic damage in burst and slows. However, if the enemy's percentage current health is equal to or higher than your own, it will stun instead. Most of the time, it's better to save this skill during trades until the enemy gets your health lower than theirs. Equilibrium Strike is a big part of why you're so strong when both you and your opponent are low HP.

Your ult is Transcendent Blades. It gets a lot of flak for being a "weak" ult, but it's actually super strong for you. Each time you activate it, it will send out a blade in a line dealing damage and healing you for a % of that damage, up to a total of 4 blades. The damage isn't bad and scales quite well, it heals you for a good amount, and provides good wave clear if you need it. If the enemy has a large minion wave, you can actually still fight them as long as Transcendent Blades is up- it will heal you more the more minions it hits. Its most important function, however, is getting you lots and lots of Trinity Force procs. If you spread out the actives, you can get 4 bonus Trinity Force procs from it, which is just an insane amount of damage.

Irelia is a very good roamer. You should always be on the lookout for opportunities to gank mid or Teleport bot. In mid, you ideally just walk down to their lane, come in behind them, Equilibrium Strike, Hiten Style and auto a few times. With the damage from your mid laner, they should be dead by the time you Bladesurge them.