This is my first guide that I have made, but this build I use can be used many fighters. I use it with xin and udyr when I jungle. However, it makes more sense to lane with Irlelia since she is one of the best farmers in the game. Always ask for a solo lane if you have a jungler.

Why I build her this way: To counter the meta game.
Full tank teams can just towerdive the opposing team without a single death, and if they have a protected healer, its gg.
Madreds is one of the only anti-tank items in the game, but I feel not enough players get it even when the opposing team has three or more tanky champions.
The most asked question is why dont I max my attack speed with this or get any lifesteal.
Health regen should never be a problem with your ult and heal.
As for maxing attack speed, I have a question to answer the one I awalys get.
How much damage are you doing when you are dead?

For fighter Irelia-
I prefer Attack Speed reds, which increase Madreds and Hiten Sytle dps.
I suggest for yellows mp5, but if you are getting clarity/teleport or just feel you can make due without the extra mana, get flat armor to live longer.
Most damage in the game comes from spells, especially aoe late game. This is why I like the extra magic resist for blues.
Since Irelia is highly item dependent, I like gold/sec quints, just for extra gold for wards, or for pots if you are having trouble laning.

As building as a fighter champion, Irelia makes good use of sligtly increased health regen, less damage taken, and one gold per ten seconds. I get improved ghost and heal since those are the summoner spells I usually get.

Laning:
-Ability to get many minon kills with bladesurge, then waves with your ult at lvl 6.
-Can easily get kills if you can harass them(your regen should allow this).
-Needs to farm, cannot be denyed(GP or tough 2v1).
-I can often deny my opponent exp and minon kills, but watch out you don't push out far if they have a jungler.

Jungling:
-Less likely to feed at low levels.
-Slowest jungler I have ever played, even if you start at blue.
-I would jungle only if your team requires one.
-Almost any champion can gank early with red buff, just wait untill they are pushed out far enough.

Irelia always needs ghost or flash, as she has no natural way to flee(however I tend to use ghost mainly to chase).
For her second spell, ignite is very popular, as it helps to finish off champions out of mellee range, especially when used in combination with Transcendent Blades. I get Heal so I can towerdive early on and live, but if the opposing team has one or more ad carries, exaust will be much more useful.

I max bladesurge for the easy farm, especially since its mana cost does not increase with rank. After that, I get one rank of Hiten Style for more regen if they are harassing, and start to max ranks once I get Madreds. I usually max Equilibrium Strike second for the nice burst damage and longer stun duration(you were tower diving who?).

In my opinon, Regrowth Pendant is the most op starting item, as it doubles health regen at lvl one. This item really allows you to easily stay in your lane until level 9+ at which point you should be ganking or just helping other lanes. I use my mana pot usually just before I want to dive or kill someone in my lane, but if its high elo, use it to farm more. I rush heart of gold to get the the extra gold per ten seconds, along with the nice stats of health and armor. I get frozen mallet next since early on madreds does not provide that much damage, and mid-game its slow is a must for chasing. After I get madreds, I stack defenses, depending on how much damage the other team does with ad or ap. This is just my standard purchases, Spirt Visiage for the magic resist and increased health regen, and force of nature for the regen, movement speed for chasing, and large amount of magic resit. Randuin's Omen is great for teamfights(when either team is attempting to feel, or just against ad carries) and it provides alot of armor, so if your score is possive by 2 or more kills, you should rush it(after madreds) in order to tower dive for longer.

Farming is easy unless you are harassed alot. Just auto attack to get minon kills, and use bladesurge to get minon kills farther away, or if your attack speed isn't high enough to get two at almost the same time. Your ult use is very imporant. If you are harassing whoever is in your lane alot(they are around 1/2 almost all the time) save your ult to finish them off. If you are they one at 1/2 hp alot, use it to heal yourself, expecially if and when you get ganked. Else just use it to farm minon waves.

Irelia, using this build, is very viable for competive play. For teamfights go first for a healer if they have one, they always must die first. Then carries, and lastly tanks. Irelia is fun to play, but I personally like playing Akali more, but can't when they other team stacks tanks or cc, which is what ranked games are all about. If you are getting focused in team fights, you are doing your job, just buy a guardian angel. Remeber you are not playing by yourself contribute to your team with wards and aid in lanes and teamfights.
I would show you my past games, but LoL is down right now, which is why I wrote this build(+basic guide).

Play Video Teammate: "HotshotGG your going to get ganked"
Xin Zhao: "A Demanican does not Retreat"
Announcer: "Pentakill"