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.

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.

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.