The "not sure if trolling or just stupid" meme comes to my mind. Well, your build is kind of alternative but not an option to me. In most of the games you'll be a dead meat before reaching force of nature and frozen heart. You have barely no survivability and to deal damage as Irelia you need to directly enter battle. You accurately mentioned in 'cons' that she is easily focused. The conclusion is easy - You die before you do anything to anyone.

Considering the items you have chosen (I am not going say what to buy - it's not my build - just some critic about it):
Trinity Force - still not a bad option but not as viable as it was before nerfs on her Q and E, moreover expensive and postponing further build.
Bloodthrister - As the statistics in the beggining say your attack speed with full build is 31% - how the hell you want to use the life steal attacking once per second?
Frozen Heart - you pay for 500 mana? Irelia does NOT lack mana. Buying rendurins is the only option. Comparable armor, health, health regen, active slow. RO >> FH
Infinity Edge - what the ****? how does it sygnerize with anything you have?

Assuming the total cost with some wards and some elixirs of fortitude/brilliance you'd like to buy somewhere inbetween, the total build cost is around 21k gold! How long you want to play to get it? 1,5h? No. During the 50 minute game you can achieve 16-16,5k. I've played over 300 games with her and I know how it goes. And this means you actually give up 2 of your last items as you won't have the money for them no matter what.

TL;DR -> You wanna die as ****, try this guide.
OFC -1.