7 kills is for PvE not PvP. I was stuck on that part for a long time. I actually managed to get a 7 kill streak in crucible with it too but it never checked off on the quest line. At first I thought ok maybe I have to get 7 kills with no assists in between (managed that too) still no progress. Went into PvE killed 7 opponents and boom quest completed....well at least until the next reset. Wish bungie would clarify.
Or try Control. The game lasts longer so you have more time to get headshots. And the opposing team will always be heading to 1 of 3 zones. Just lurk outside radar edge and ambush. I'd use grenades or smoke to lower health then hope the last shot was precise. It took me about 15 matches before I got it. Oh and play during off periods - before school lets out, not on weekends, etc. It's hard to compete with 15 year old hand eye coordination.
Wait you're 15? I did it in rumble because most fights are 1v1 so it's easier to kill the person with a headshot. I got it first try on a rumble match after trying it 1 time in control.