AP Irelia used to be somewhat viable, but then they nerfed her Equilibrium Strike scaling from 1 to 0.5. Don't bother stacking ap. TriForce synergizes well with her.
Her ultimate is her only ranged attack, good for catching someone with low health if your dash is on cooldown. It's great for team fights as you can just send it through everyone and heal yourself up.
As for her build, I usually start with a regrowth pendant, 1 health pot. If it's a heavy ad top (like Tryn or Renekton), I go cloth, 5 health pot. Rush a Philo stone, then get either Merc Treads or Ninja Tabi (situational). After that, get your sheen then Triforce. That is your corebuild. I usually get a warmogs after triforce, but there are other options to go for too.
Incase you're wondering about skill order, I get my Q at level 1 just for easier farming, then I max out W, getting E at level 4. After maxing out W (and getting R at 6, 11, 16), it really depends on play style and situation. I like to max out my Q second so the cool down isn't so long when I go into team fights, but maxing out E is better if you're doing a lot of 1v1 or if you need to cc someone hard.
To sum up everything, don't build Irelia AP, build her tanky dps. Ult heals and is a ranged attack. Rush a Philo, then get Merc Treads/Ninja Tabi, then get Sheen ---->Triforce. Get Q at level 1, then max W, getting E at level 4. Max E or Q second, depending on your play style.