Once an Assassin creature or commander you control deals combat damage to a player in a turn, you can cast spells for their freerunning costs for the rest of that turn. It doesn’t matter what happens to that Assassin or commander after that.

If the target creature is a legal target but isn’t destroyed, most likely because it has indestructible, you’ll still draw a card if another creature died this turn.

If the target creature is an illegal target by the time Chain Assassination tries to resolve, it won’t resolve and none of its effects will happen. You won’t draw a card even if another creature has died this turn.