If I use Baited Hook on a Ruined Rex and summon a Karthus, does the Deathknell effect from Ruined Rex happen twice?

No, the Deathknell effect from Ruined Rex will only trigger one time.

According to the established game rules for Baited Hook and Karthus, Eternal, Karthus must already be on the battlefield at the moment a Deathknell trigger is created for his passive ability to apply and double it.

Sequence of Resolution:

1. Baited Hook resolves: You kill your Ruined Rex.
2. The Deathknell trigger for Ruined Rex becomes pending. It cannot enter the chain yet because Baited Hook is still in the middle of resolving its instructions (Rule 734.1.d.2).
3. Baited Hook continues: You look at the top 5 cards and find Karthus, Eternal. You play him, and he becomes a pending item on the chain.
4. Baited Hook finishes resolving. Now, all pending items finalize in the order they were created:
   • First, the Deathknell trigger for Ruined Rex finalizes and enters the chain.
   • Second, Karthus, Eternal finalizes and enters the battlefield.

Because the Deathknell trigger was created and placed on the chain before Karthus was on the battlefield, his passive ability ("Your [Deathknell] effects trigger an additional time") does not see the event to double it.

This is based on RiftJudge FAQ #1116, FAQ #9256, and FAQ #8551, which all confirm that Karthus must be present when the trigger is created.