First off, hide the objective you want hidden with ["1" objstatus = "HIDDEN"]. A nice big trigger will now do the rest. If the unit you want killed is called "badguy", then put in the Condition field of your new trigger [getdammage badguy = 1]. In the on activation put ["0" objstatus = "DONE";"1" objstatus = "ACTIVE"].
Note: 0 and 1 are whatever number the objective is in the list of objectives in briefing.html. Change them if appropriate. Also, the brackets are for seperation. Don't use 'em in the fields.