Here's some tips on how to finish Jolly Holiday on Destiny: The Taken King.
Tip #1: Tilt means I stop.
This bounty punishes you in a way that nothing else does. So it's especially important to have the right mindset for this particular goal. If you feel you're not in the mood of doing it, then don't do it.
Tip #2: Ignore the match objectives, score, teammates. Just survive.
-It's dirty, selfish, and awful to play this way. But in the context of this bounty, you're the one who stands to lose way more than anyone else from deaths. Play slow, play sneaky, play however you need. As long as you're not dying, and until this part of the bounty is over, stay alive however you can.
Tip #4: Rift is your best friend.
There is only Rift. All hail Rift. (Or Mayhem, according to a lot of people here)
-Rift has elements of game play entirely unique to PvP modes. Make use of the enemy focusing their attention on the spark. Flank the enemy when they're waiting for the spark if you can. Tripmines work great for where the spark will charge. Never pick the spark up, let your teammates have the glory. You're there to get stragglers and anyone who has a red bar over their head. If the other team gets the spark, get back near your rift IMMEDIATELY and take up position to get kills on the one or two that may get through. But keep a good distance from the rift in case they ignite it, otherwise you'll die horribly.
Lot of people had luck in Mayhem. I have no idea how (because Mayhem is absolute chaos), but apparently that works great for some people. To everyone who suggested that, I tip my hat to you, you crazy, crazy people.
Tip #5: Radar, Radar, RADAR
Watch it nonstop. 'Nuff said.
Tip #6: Positioning cannot be understated.
If you're keeping good watch on your radar, you're going to know if you should run or hide. Those are your two options. Don't be afraid to run. You're there to kill people who don't see you or know you're there. The range on TLW is really bad, so you have to use every advantage you can. The layout of the level, where they are going to come from, how much cover you have, escape routes...it all factors into fight or flight during this step. Generally, try to stay behind anyone who is near you. Your chances for survival go up dramatically if you keep teammates between you and the other team.
Tip #7: "Destiny Math" is confusing, so find what works every time without any uncertainty.
-Special weapons (fusion rifles, sniper rifles, etc) and heavy weapons (Swords, LMG's, rockets) are questionable at best. Countless anecdotal evidence (including mine) call a lot of things into question on whether or not the kills count or how much do they count for. There is a lot of confusion about this. I don't use any other weapons for this bounty anymore, unless you get ambushed or caught off guard, in which case you'll use a shotgun just to stay alive. Grenades, melee, and supers are pretty much money in the bank, however, and quite frankly helped a LOT to get me to 100%.
Tip #8: Swords? GET OUT NOW...
-If the other team is packing swords on more than one or two people, don't even bother. The odds are never going to be in your favor. Check the other team in the roster menu, if you see swords everywhere then you need to put Last Word away for now. In a match up between a sword and TLW, sword wins more often than not. Unless you're magical and godlike with TLW in PvP, you don't stand much of a chance. Besides, this guide is for non-magical and non-godlike people.
Tip #9: Always equip The Last Word right before victory
-Even if the match has not been favorable for you to use TLW in any capacity, if it's the final two minutes or the score limit is very close to being hit, and your team is winning then get to a safe spot far away from everyone else and equip The Last Word. You get progress just from being on the winning team with it equipped.
And of course, if you finish the Jolly Holiday Quest, your reward is this: