When I have Darius Trifarian and my opponent has a Vex on the battlefield, I played my Darius after killing the Vex, but I played the Darius Trifarian as the fifth card on my turn, so does my Darius enter ready, or does he need to be the second card or see the second card played in order to be ready?

No, Darius enters exhausted. His ability only triggers when he is exactly the second card played in your turn. Playing him as the fifth card means the condition ("second card") has already passed — he won't see it and won't ready himself.

This is consistent across multiple rulings (FAQ #3905, #7612, #9420). Darius can count himself as the second card if he is the second card, but as the fifth card he's past that window entirely. The fact that you killed Vex first is irrelevant to his trigger condition — only the card order matters.