First off let me begin to say that this guide for Irelia is at its most effective when used in 3 v 3 play, however it can still be effective in 5's. That being said, as stated in the build title this is an ap nuke build for irelia so sit back and enjoy the ride.

For runes on Irelia I choose magic pen marks, and then flat ap for seals, glyphs and quints.
The reasoning behind these runes is for early game dominance. this comes from being able to harass extremely well with her E which should be your first skill pick, this Ability starts with a base of 80 so ur 1 lvl output will be 80+50 = 130 from your first item, the amplifying tome. and as this build is about early game dominance this is good news for you. with 130 + a slow or stun at level 1 you will be very scary.

However if you feels these runs are a bit too glass cannon you can sub out the ability power quints for flat hp for the extra 97 health and still do fine.

in masteries i go with a 21/9 build again focusing a magic pen, ability power and early game damage while also getting the small dodge buff from the defensive tree.

Items for this build like most can be very situational, i will list the items in order for the perfect game, and then items for a game with safers, or a game where it is a dead lock so little kills are had and farming creeps is the focus

PERFECT game
amplifying tome --> sheen
boots --> ionian boots of lucidity
amplifying tome --> mejai's soulstealer
blasting wand ---> rabadon's deathcap
blasting wand --> voidstaff
hextec revolver --> gunblade

this may seem like an expensive build, but when ur super fed from owning early game it is easily achieved unless they /sur which is the general case.

Unperfect game
sheen
ionian boots
rylias
void/abyssal scepter or both (if focus'd or if they have heavy ap)
hourglass
gunblade

again an expensive build late game but Irelia is a very good farmer using her q to last hit minons is a necessity.

your E is going to be your main source of burst along with your ULT (r)
using this thought process you should level as follows

1-E - initial damage and harassment
2-W - true damage on auto attacks, healing per auto attack --> lane power
3-Q - got distance closer and intiater --> UR FARMING ABILITY USE IT TO LAST HIT MINIONS

from this point you will want to max your E first putting points into W when E isn't available always putting points into your ultimate and leveling q when its your only option.

Summers spells is pretty strait forward here

ghost - great for catching up to q then e into a slow or stun
exhaust - great for staying alive vs carry ad dps, or stopping a bursty ap char with its recent remake

optional
flash - works likes ghost as a gap closer but also good for escapes, since with this build u must be aggressive
ignite - usually not needed since you nuke anyways, however useful for champs like mundo which you see frequently in 3's

Probably the most important part of this guide, FINED a friend and LANE BOT together my personal favorite compliment to this champion is ryze, but a pantheon is also a great choice as well as a xin zhao or nidalee

The strength from this build paired with Ryze is immense early game 1-6 you will want to both harass very hard, using Irelia's E, and Ryze's E which is the ball.
during this phase, harass hard as I've said and wait for 1 to over extend have Ryze snare,then proceed to use all your abilities and obtain first blood during early game its important that you rush sheen for the damage buff.

After level 6 your damage is greatly increase as Ryze and yourself obtain your ultimates. AT this point you both should gank top if the gank fails return bottom you should be able to hold your lane champs at their tower, and if harassed enough you will have the damage to tower dive if you are at full hp as Irelia do NOT be afraid to spam your ult, it has a very low cd with the new boots.

continue this process and you will soon be fed with a 17 min surrender.