Are deathknell triggers immediately put onto the stack upon that unit hitting the grave or do they start a new chain after the current one cleans up?

Ruling: Deathknell triggers are immediately put on the chain in the cleanup after whatever caused the unit to die.

Sequence:

• When a unit with deathknell dies during chain resolution, its trigger is added to the current chain immediately
• Priority is given to add reactions to the chain
• The chain continues resolving from the top
• Example: If Hidden Blade is 4th in chain and kills Watchful Sentry, Sentry's draw effect is added to the chain before cards 3, 2, and 1 resolve

Nuances:

• Focus and priority are separate concepts: Focus is permission to start a chain during a showdown (with an action if desired), while priority is permission to add a reaction to the chain
• Attackers always get focus in a showdown
• Focus automatically passes to the next player each time a chain fully resolves