I have a Vex, Apathetic with a Spinning Axe and 4 damage marked on her and a hidden Evelynn on the battlefield. My opponent has a Karthus and his Ruined Rex which just died, has two instances of Deathknell on the chain. If I reveal my Evelynn while he has targeted Vex on the chain, can he target my Evelynn with the second instance of Ruined Rex's deathknell?

No — the second Deathknell instance's target was already locked before you could reveal Evelynn.

Both of Ruined Rex's triggers (base + Karthus doubling) become Pending at the same time, and during the Finalize step they are completed in order without passing priority (Rule 337.1.a, 337.3) — the opponent chooses each trigger's target as it finalizes (Rule 402.2), while Evelynn is still hidden. Only after all Pending items are finalized does anyone get a chance to react (Rule 337.4). So when you reveal Evelynn, the second instance is already on the chain with its target chosen, and that target can't be changed — revealing a Hidden unit in reaction never retroactively makes it a valid target for a trigger whose choice already happened (see target selection at finalization, Rule 352.4.b / FAQ #737).

Your reveal only affects anything going forward (e.g., Evelynn's own play ability); it does nothing to the second Deathknell's target. If the opponent wants the second trigger to hit Evelynn, they needed to pick her at finalization — by the time you reveal, it's too late.