User Cadenza Lunare has a great guide over on TA for this one.
For the record, unique weapons of the weapon type necessary to complete these trophies count. For example, Dinner Bell is a shotgun, and counts towards all GRA challenges that require a shotgun. Also, the Killing Blow is all that matters. You could weaken an opponent to 1-2 bars of HP with any weapon, and finish them off with the challenge weapon, and then still get credit.
Also, dismiss your partners, as they can screw you out of a challenge by taking the shot themselves.
Against All Tyrants::
***Kill ten named Legion members with Hunting Revolvers, Ranger Sequoias, Brush Guns, Service Rifles, Hunting Rifles, or Cowboy Repeaters.
PROS :: The weapons required for this challenge are fairly common, even in the late game.
CONS :: Highly limited number of named Legion members. Easy to screw yourself out of this one.
RECOMMENDED :: If you can go for this one and you have enough Legion alive (check the list below), by all means do this one. Not recommended for people attempting to do a Legion ending.
Acquiring the weapons required is fairly simple - NCR Veteran Rangers like to carry Ranger Sequoia, and you get one for free if you complete the Return to Sender quest and convince Chief Hanlon to commit suicide. I wager you could also simply kill him and take it. Otherwise, the majority of these weapons are purchasable from the Gun Runners.
The challenge will be impossible if you've already undertaken NCR quests which require you to kill Legion members or just gone on a killing spree against the Legion. For me, this challenge was impossible because I had already killed half of them just doing quests.
For the rest of you, there are only 17 named Legion members with names in the game, one of which is the final boss, and Alerio only appears if you killed Vulpes Inculta in Nipton. You can get credit for Vulpes Inculta AND Alerio, if you kill Vulpes in Nipton with a required weapon, and then kill Alerio when he shows up later, upon retrieving the Platinum Chip. Silus, the Legionaire held hostage in Camp McCarran, does not count because you cannot bring the required weapons inside his cell. Here are the valid targets' identities and locations ::
Captain Ronald Curtis - Camp McCarran (Insider for the monorail bomb quest, disappears if completed)
Dead Sea - Nelson (Inside a Barracks in Nelson)
Atticus - Caesar's Legion Safehouse (Only accessible if you are good enough with the Legion for the key)
Aurelius of Phoenix - Cottonwood Cove (Second floor of the HQ building)
Canyon Runner - Cottonwood Cove (Patrols the area outside)
Cursor Lucullus - Cottonwood Cove (The ferryman, moves to the Fort later)
Decanus Severus - Cottonwood Cove (Patrols the area outside)
Alerio - The Strip, Outside the Tops (Only shows up if you killed Vulpes, otherwise Vulpes does)
Gabban - Nipton (One of Vulpes' men)
Vulpes Inculta - Nipton/The Strip/The Fort (Kill Vulpes at Nipton to get Alerio to spawn later)
Karl - Red Rock Canyon (The Legion emissary at RRC, sleeps in the Longhouse. Can be killed by Papa Khan and friends in the Oh, My Papa quest)
Alexus - Techatticup Mine (Inside the Mine)
Antony, Caesar, Lucius, Otho - The Fort (Antony is inside the Fort, outside Caesar's tent. Caesar and Lucius are inside Caesar's tent. Otho is inside the Fort, by the arena.)
Legate Lanius - Final Boss (Non-Legion Endings)
As you can see, it's very easy to screw yourself out of this one if you, like me, didn't expect the challenges. However, in theory this is one of the easier ones to execute if it's available to you.
Armed For Bear::
***Kill 15 named NCR Rangers or commissioned officer (Lieutenant, Captain, Major, Colonel, General) with Machetes, Rippers, Chainsaws, Throwing Spears, or Ballistic Fists.)
PROS :: Since most people don't usually have a reason to kill these people, they're usually still available to most players. The weapons are easy to find. There are about 35 named NCR personnel.
CONS :: If you killed all of NCR, you're screwed out of this one.
RECOMMENDED :: If you have the NCR personnel alive, do this one. If you're good with Sneak Attack Criticals, even NCR ending characters can get this challenge done.
If you don't want to lose reputation with the NCR, you can pull off sneak attack criticals on these and escape with no loss to reputation. Machetes and Ballistic Fists can be found from Prime Legionairies and other Legion personnel in the Fort, and Ballistic Fists are incredible for Sneak Attack Critical kills even on low strength/melee skill plays. Otherwise, the Gun Runners tend to sell Rippers/Chainsaws or you can get a Chainsaw from Motor-Runner in Vault 3.
If you go to the Ranger Stations Alpha/Bravo/Charlie/Delta/Echo/Foxtrot, each has a Comm Officer and a named Ranger which you can assassinate. The only exception is Charlie, which only has one Comm Officer and he can get killed in a Legion Raid late game. If you kill all the named Ranger Station personnel, that is 11 towards your total, and you only need to kill four more. If Camp McCarran is still intact, you can find five named personnel there, unless you completed the Three-Card Bounty quest which will cause First Recon to move to Camp Forlorn Hope. Camp Forlorn Hope has Major Polatli in the main tent and the First Recon officers if they have moved there. Bitter Springs has 2 named officers, Helios has 1 standing outside the main entrance, and Novac has old Ranger Andy in one of the hotel bungalows. The Mojave Outpost also has three named personnel - two in the offices, one on the roof of the bar/barracks where you recruit Cass.
***Kill ten Mojave Wasteland Cazadors with Tin Grenades, Long Fuse Dynamite, and Time Bombs.
PROS :: Cazadors can be fought relatively common, and on demand at The Thorn.
CONS :: You have to fight Cazadors. The weapon selection is highly prohibitive and hard to get in the late game, or get specific perks.
RECOMMENDED :: Only if you just happen to have the means to get the weapons or have them already. Bring antivenoms, just in case.
This one is a pain, and I only recommend doing this one if you have absolutely no other choice. Tin Grenades require the Mad Bomber perk, 25 explosives skill, 1 tin can, 50 pistol powder, and 1 duct tape. Long-Fuse Dynamite is hard to find late game if you already savaged the Powder Gangers, and Time Bombs require 50 Explosives skill, 5 dynamite, 1 egg timer, 1 duct tape, and 1 scrap electronics.
If you've already screwed yourself out of the NCR and Legion killing challenges, you need to get AT LEAST 10 of either of these weapons, and find Cazadors. They can commonly be found in the vicinity of Jacobstown/Remnants Bunker, east of Helios ONE, or north of Goodsprings near the Powder Gangers' vault.
However, the best way to prevent a long search is to do the quests for Red Lucy at the Thorn until you're able to request the ability to fight Cazadors on demand.
If you do this, you can weaken the Cazador before the gate even opens and finish them off in VATS with the explosive, and repeat as desired. It is highly recommended that if you are going for this one to save before an attempt and after a successful one as to avoid having to farm up more of the explosives.
Deathclaw Pro Hunter
***Kill 5 adult Mojave Wasteland Deathclaws with .22 Pistols, Switchblades, Boxing Tape, Recharger Rifles, or Dynamite.
PROS :: Deathclaws can be fought on demand at the Thorn.
CONS :: These weapons are a pain to find in the late game. Deathclaws tend to tear you apart.
RECOMMENDED :: If you know what you're doing, this one can be a cinch, especially if you have Deathclaws to fight at the Thorn.
If you're going to be doing this one, by far the best way is to unlock the Deathclaw fight at the Thorn by completing Red Lucy's quests, and fighting them one on one. You can land sneak attack criticals with your normal weapons before the gate even opens, get them to low HP, and then finish them off with the weak weapon in VATS. Use whichever weapon out of the base list your character is most proficient with to prevent any unfortunate accidents.
If for whatever reason you can't go to the Thorn due to killing them all, Deathclaws spawn around the Quarry Junction, the Gypsum Train Yard, and south of Vault 3 near the mountains. However, it is much easier to fight them in a controlled environment.
***Kill 10 Mojave Wasteland Mr. Gutsys and Sentry Bots with Brass Knuckles, Spiked Knuckles, Tire Irons, and Lead Pipes.
PROS :: Robots are on a respawn timer at New Vegas Steel. The weapons are easy to find even at high level. Love and Hate count as Brass Knuckles.
CONS :: None, really.
RECOMMENDED :: If you can do this one, do this one.
This one is a breeze - first and foremost, there are a good amount of Sentries and Gutsies at the Crashed Vertibird east of the nuclear test site. If you need a pipe or knuckles, go fight some Fiends near Vault 3 and you're bound to come across some who are holding one.
Go to the Crashed Vertibird and weaken as many as you can, then go nuts in VATS swinging to finish them off with your melee weapon of choice. If you need more, go to New Vegas Steel, near Camp McCarran. Inside are three Mr. Steel, that count as Mr. Gutsy. They are on a three-day respawn timer, so use them to kill off any for your total remaining, and go to Goodsprings. Wait for 24 hours, 3 times, for a total of 72 hours, and return until you have all you need. If you are not strong enough to take on the Crashed Vertibird robots, you can also take some out in the REPCONN Headquarters. If you manage to make it to the top and shut down robot security, you can just go back for all the Gutsy inside while they are disabled and hit them until they explode, unable to defend themselves.