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One Spartan Army - Kill 400 enemies in Firefight Matchmaking. - 2000cR
Four-hundred kills is nothing to bat an eye at. Normally, a Firefight Matchmaking game brings in a hundred and twenty kills. This means you will be playing at least four games. For the fourth game, I would recommend playing Crashsite. Once you hit the required total, just let the generator explode or die. Gruntpocalypse is a very easy way to get kills but I will also encourage players to play co-op due to the Weekly Challenge.
That Pink Mist - Kill 70 enemies with a Supercombine from the Needler or Needle Rifle in Firefight Matchmaking. - 1000cR
The Firefight playlist has changed some since we last had this challenge. Players can no longer pick up weapons in Sniper, Rocket, Grenade, Fist, or FRG Attack/Fight. This means that you will want to aim for FRG Attack since your starting weapons are the Fuel Rod Gun and the Needler.
Now, this will probably take several games. The only waves you can really use the Needler on effectively are the Brutes and the Elites. The Grunts will die before the Supercombine explodes and the Skirmishers/Jackals either have the shields or are too agile. Do what you can in those waves and grind it out while getting the four-hundred kills for "One Spartan Army."
Neuroscience - Kill 10 enemies with headshots in a multiplayer Matchmaking game. - 1200cR
It figures that, today, they would force you to also play multiplayer to get all the challenges. My advice for this challenge is SWAT, Snipers, or Team Objective. Any of these can net you ten headshots easily. Remember for Team Objective, if you cannot find a DMR, you can use the Magnum, or cut through their shields with the AR and the switch to the Magnum for the headshot.
Five-time Champ - Win 5 games in multiplayer Matchmaking. - 1750cR
Between the four games of Firefight you will have to play today, this challenge will probably end up being a pain. It is requiring five wins, not just five games played. This is all up to personal preference. Play whatever gametype you play best in. You should personally know what gametypes you tend to win in, so stick to those and you should be fine.
Endure: The Return - Complete four sets on any Firefight Map in co-op play on Heroic or harder. - 10000cR
Now this is an interesting challenge. Four sets is quite a long match. A set is composed of three rounds (four if you include the bonus), and a round is composed of five waves. As you can tell this is multiplying to quite a bit. If you include the bonus rounds, you are talking about sixty-four waves of enemies. Doing this Co-op makes that much easier, at least. Now, nothing on this challenge says it needs to be done in a single game, but I am going to assume that it will need to be done in a single game. It just seems like something Bungie would do.
As well, the challenge says nothing about Matchmaking and, indeed, the only place this might be possible is Firefight Limited, but you would be fighting the clock. And if the game is offline, that means you have full control. You can make yourself invincible, make all the waves Grunts, get unlimited ammo, change the skull effects for each round and set. There really is a lot of possibilities and that makes this challenge much less daunting. You can even just turn on a second controller and play by yourself. And a large amount of respect to any players who do this the hard way, either in Firefight Limited or in a standard game of Firefight.
If you have any helpful tips such as maps or gametypes, post them below. Once I get back from work, I'll put together a more in-depth guide and maybe a video. But the video will not be up till tomorrow at the earliest.
Edit: Two of our readers have you covered! Check the first three comments below from Greg and Matt for some really easy ways to get the Weekly challenge done.