Irelia is by far my favorite champion. She is a monster nuke with solid survivability and can lane/last hit like a champ. She can carry a donkey of a team with just a little bit of strategy and knowledge of the game. Currently she is EXTREMELY underrated but I'm not sure how long that will last.
Here are some of the ups and downs to playing Irelia:

Pros:
- Early Game Utility
- Early Game Nuke
- Mid Game Nuke/Carry
- Late Game Carry/Offtank
- Plain and Simple Complete Beast
- Short Ulti Cool Down
- Fantastic Stun/Slow
Cons:
- Damage Starts Curve off end game
- Vulnerable to physical DPS (i.e. Tryndamere and Master Yi)

NOTE: THIS IS IN NO PARTICULAR ORDER. FOLLOW THE ONE BY THE SKILL ORDER

The Mercury Treads give you great CC reduction and resistances early game. They are also good to have because of how vulnerable Irelia is to CC. You could go swift boots but the Trinity Force gives her a good amount of speed later on.
The Trinity Force is one of the best, if not THE BEST item for Irelia. The sheen proc combined with the slow is such a great combo. The fact that it also provides health, mana, AP, AD, AS, AND Crit chance, is too perfect for her as a minimal value of each is what she needs early. Try to time your procs so you get 2 procs off of your ultimate, because yes, each dagger you throw has the potential to provide a sheen proc for 120% of your AD.
The Randuin's Omen is also a VERY strong item for Irelia. The armor is a huge plus to her squishiness. Also, the health is a great addition to the current tanky metagame. Lastly, the slow that this can provide could be just the thing your team needs. The fact that you have a flash skill can get you into team fights providing AoE slows.

Now for the fighting sequence. Irelia is a great initiator when she has slightly less health % than her target since her equilibrium strike stuns when she is lower hp % and slows if higher.
It is a good idea to keep your health just about 90% at all times so your equilibrium strike stuns instead of slows after bladesurging to close the distance.
so here is the order for 1v1 ganks: W-->Q-->E-->R-R-R-->and repeat. Don't be afraid to pop ultis left and right since your cooldown is only about 25 seconds.
team fights is a different story. try to let your teammates initiate and come around back to destroy those pesky casters. DO NOT JOIN TEAM FIGHTS STARTING WITH YOUR BLADESURGE! hit the active on your omen to slow enemies running away as well as slow their attack speed. use bladesurge to get in that last hit on someone multiple people are targeting or to escape. surging towards their creep, even if its in the wrong direction, isnt necessarily a bad idea. use it to get out of the middle without wasting the time of running into people and getting blocked. When using your ulti in a team fight make sure it is hitting as many people as possible, remember that everyone you hit with any attack is healing you for 26 health a pop.

EARLY GAME:
- Try to lane with a strong nuker or damage dealer (garen, leblanc (if she is stupid enough to lane), poppy, shaco). While laning with them, try to have your health JUST under full, via creep kill or taking some aggro. You can heal this back anyway with lifesteal and ult and it lets you get your stuns off early, and a 2-300 dmg stun at level 5-7 is insane.

MID GAME
- Start ganking lanes, using your Q > E combo to land stuns for your allies to destroy the other team, and finishing people off with your R. Also, start grabbing lizard from jungle, just to get that extra edge. Golem buff is also nice for cooldowns.

LATE GAME
- In late game 5man ganks, your job is SOLELY to take out their AD or AP carry. You can't deal significant damage to most tanks, do what you are made to do and you will succeed. Stunning their carry's can totally change the tides of a fight, as that 2 second stun saves your team of 2-3k Damage against you, and takes out one of their heroes.

Alright that's about all i have the patience for so im done.
If you have any questions please comment, I'd Be HAPPY to answer them. please vote so i can get some feedback.