I baited hook a Glasc Mixologist. Baited Hook finds and plays a Karthus. Will the Karthus that was just played through Baited Hook trigger the Glasc Mixologist Deathknell to trigger an additional time?

No, the Karthus, Eternal played via Baited Hook will not cause the Glasc Mixologist's Deathknell to trigger an additional time.

According to RiftJudge FAQ #9256 and Rule 808.1.d.2, Karthus, Eternal must already be on the battlefield at the moment the Deathknell trigger is created for his passive ability ("Your Deathknell effects trigger an additional time") to apply.

Sequence of Resolution:

1. Baited Hook begins resolving and kills your Glasc Mixologist.
2. The Deathknell trigger for Glasc Mixologist is created and becomes a pending item. Because Baited Hook is still in the middle of resolving, the trigger cannot enter the chain yet (Rule 808.1.d.2).
3. Baited Hook continues resolving: you look at the top 5 cards, find Karthus, Eternal, and play him. 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 Glasc Mixologist 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 board, his passive ability does not see the event to double it. You will receive only the single, base Deathknell trigger.