Irelia is a solid counter-carry. Her primary role is to shut down and threaten squishies in the back with her insane mobility and burst. She is much less gimmicky then Akali or Poppy, two other champions that share the same role as Irelia (counter-carry, mobility, burst). Meaning if you enjoy playing Akali and Poppy, you will see much greater success playing Irelia.

Your primary goal is to farm. You will find yourself last hitting at tower often, so be comfortable doing so.

When you reach level 6, clear the wave with your ultimate and buy items(get a ward!). Since Irelia is such a strong duelist, I recommend farming your lane for at least the entire duration of early game, scoring a kill whenever possible, otherwise harassing them out of lane with Bladesurge and Equilibrium Strike, or perhaps just farming passively.

Irelia's mid game is generally still just farming top very effectively, at this point you should have 1-2 gold/5 items, just sit on those and cs as well as possible. If your lane partner leaves top to pressure dragon, either follow or kill their tower, and make sure your team doesn't engage them at dragon if you choose to do the latter.

Irelia is insanely strong once she gets 5-6 items, she becomes near unkillable. With FoN and Phantomdancer's movement speed, Irelia moves at near 500 MS. At this point, Irelia should be actively engaging in fights and initiating for her team. Choose your fights wisely, take advantage of your deceptive tankyness and burst.

Remember your role is of an assassin, however your survivability matchs that of a bruiser. Push Irelia to her limits by maximizing her insane survivability in teamfights and try and stay in the fight as long as possible. Irelia's greatest strength is her presence in a teamfight, knowing how to abuse this presence and fully utilizing her passive and self healing to the extreme is what every Irelia player should strive for.