If you're doing the quest as a hunter, for the seven kill sprees you will want to go as a gunslinger using chain of woe. Go into the mission where ghost says "we've woken the hive" and then get precision kills on the thrall until your chain of whoa stack is maxed. Even if you do not activate the triple tap perk of the gun in order to get that 7th kill in time, the chain of woe stacked will make your reload speed fast enough where you can get the 7th kill quickly.