HI! I am a Gold rated player for 4 season in a row, i have mained Irelia since late 2011, so about 5 years. I have seen all incarnations of Irelia, i have tried irelia in every roll but ADC, and found top and mid to be the best.
This guide is a very in-depth look into how Irelia works, Chapter 1 will be when her abilities will do, her combos, positioning, etc. basically Beginner stuff. Chapter 2 will be in-depth match-ups in lane that give me a hard time over the years(i.e Darius, Olaf, GOOD Yasuo's,).
Chapter 3 will be on Bans, and if you are running with a team or Duo, I can give suggestions on the best team comp situation to have for Irelia.
Chapter 4 will be on tips.

Olaf- He might as well be the Anti-Christ. Heres what you do. take Corruption pot OR Cloth Armor and 4 pots at the start, Get your W at level 1 and be in for the long haul, at level 2 get Q and see if you can 1v1 him.... no? then sit under tower and wait for a gank, DO NOT GIVE HIM ANY KILLS EVEN IN TRADES HE WILL OUT SCALE YOU EASY!!!!!! if yes, kill him and push your advantage, at level 6 roam, no point in fighting him anymore.
Darius- If Olaf is the Anti-Christ, Darius will be Satan himself, IF you find yourself hugging him, stay there until his Q is done, that way you dont heal him, remember kids, just the Tip... Heals him. at level 6 depending on builds if you have a sheen you win trades dont let him poke you to get Hem-stacks. he will win trades if you let him.
Good Yasuo's- You still win level 6 if you play smart, let him dash every where and wait for him to be done, then Q him, dont let him poke you with his Q, its fine if he knocks you up and ult you, as long as he does it with you at full HP you should be fine.

Bans: Olaf, Yasuo, Zed, Eve, Shaco, Vayne, Draven, Graves.
Best possible Team comp for 5 man queues
Top(if youre mid irelia): Rumble
Mid: Oriana/Annie
Jungle: Hec, Nocturne, Xin.
ADC: Miss Fortune, Jinx, Vayne
Support: Blitz, Lux, Bard, Braum.
Basically, youre using all of your ults together to create a state of confusion, most players would be overwhelmed with everything that you can slip into the back line easy.
Best possible duo partner per position
Top(if youre mid irelia): Darius, Wukong.
Mid: Ahri, Annie, Orianna.
Jungle: Xin, J4, Hec, Shyvana, eve.
ADC: Miss Fortune, Vayne.
Support: braum, Sona, Soraka, Blitz.

Farming: Don't over use your Q to farm, just because it resets doesn't mean you over use it, last hit like normal, you'll run OOM way too quick.
Farming against passive players: freeze the lane, if they try to farm or get XP, Dash to a low caster then the enemy champion and attack them, then go back to freezing the lane, Ward river to guard against ganks.
Farming against Aggro Players: Pre-6. play passively, you have a huge power spike alot of people forget about because of her weak pre-6. but dont be a little *****, play smart, tell your jungler that they're being uber aggro, easy kills, easy win.
Teamfights: Do not start fights, do not dive under towers in a 5v1(obviously), if you have a good team comp, then enter the team fight AFTER people used their ults, Irelia Flourishes on gimped players. go for the highest threat first, if you have Courage of Colossus then you should be murking people.
Summoner Spells: Usually take Teleport, and use it wisely, apply pressure to both sides of the map, or one then join a team fight. You can run Ignite for early kills. Barrier for playing passive(works great with Courage of colossus.