So let's say you are playing with Exhaust and you kill Karthus while he is Exhausted. Will his damage be reduced by Exhaust in his passive if you killed him while exhaust was active?
No. When Karthus dies and his passive is activated, all CC is removed and he is untargetable for the duration of the effect.
"Upon dying, Karthus enters a spirit form and becomes invulnerable and untargetable, allowing him to continue casting spells at no cost for 7 seconds." -From the League of Legends Wiki.