When Darius is played as a second card, does he ready himself from this, or does he need to see a second card being played?

Yes, Darius readies himself when played as the second card of the turn.

According to RiftJudge FAQ #6776 and #7647, Darius counts himself as the second card played. Because a card is considered "played" once it has fully resolved and entered the board, Darius is already on the battlefield when his "When you play your second card" ability checks for its trigger condition. Since he is present on the board at that moment, he sees himself as the second card played and his ability triggers, granting him +2 Might and readying him.

Sequence of events:

1. You play your first card of the turn.
2. You play Darius as your second card.
3. Darius resolves and enters the battlefield.
4. Darius's triggered ability checks the board state, sees that he is the second card played this turn, and triggers.
5. Darius receives +2 Might and readies himself.

As noted in Rule 402.1.c, if a unit is instructed to be readied while it is already ready, nothing additional happens, but in this scenario, Darius enters play exhausted (as is standard for units) and is then readied by his own ability.