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Eye for an Eye - Introduction
You can start this quest by talking to Sergeant Astor. He usually can be found at his base camp by the ruins of Camp Searchlight. If he's not at the camp, then he's probably patrolling the perimeter of the town.
Once you find Cottonwood Cove, you should be able to talk to Astor about it. If you bring it up, he should mention wanting revenge against the Legion for irradiating the town.
For starters, you have the option to bug their radio and steal some reports on Legion troop movements.
This is probably the best option for neutral players. It's going to be effectively impossible for an enemy of the Legion to get to the radio and the report without killing everyone in Cottonwood Cove. Note that you can just skip this part of the quest and kill everyone. You'll just miss out on a little experience.
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Eye for an Eye - Gathering Intel
In Cottonwood Cove, there is a central two-story HQ. Go into the camp and walk into either the upstairs office or the downstairs command center. In general, the office should be clear during the day. It should be possible to walk right up and steal the file you need from a cabinet. While you're here, make sure that you get Recompense of the Fallen from his desk. This "unique" unarmed weapon is quite deadly.
The radio is downstairs. You should be able to just wait around a bit until the bulk of the officers leave for their respective areas. It's then as simple as crouching and bugging the radio and grabbing the raid intel from the cabinet next to it. You should have all the intelligence you need for the NCR.
Note that there is a pro-legion side of this one. If you want, you can talk to the leader, Aurelius of Phoenix, and gain some fake intel that you can just pass on to Astor to confuse things and end the quest early.
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Eye for an Eye - Clearing Cottonwood Cove
If you already delivered the intelligence, or can't get into the camp without a bloodbath, then launch the attack.
It's actually pretty easy, and there is a shortcut.
If you want some poetic revenge and the biggest reputation reward from the NCR, you can dump a bunch of radioactive waste into Cottonwood Cove and kill everyone in one fell swoop. There is a bit of a problem though. There's a quick quest, "Left My Heart," that allows you to free a family of slaves. If they're still in the camp, then this attack will kill them too. It's your choice. In order to take this poetic shortcut, just go to the big truck hanging out on a cliff above the Cove. You'll need fifty points in Lockpick to dump the barrels out by opening the tailgate.
If you want to clear out Cottonwood Cove in a more traditional way, just bring a Companion and attack. Bring a good rifle or a gun you're comfortable with. If you're a melee fighter, then you should be fine too. The guards at Cottonwood Cove are actually pretty poorly armed. I only saw rifles and basic melee weapons, so it shouldn't be hard to pick off snipers and gun down rushers. Do watch out for any dogs though. Legion mongrels can dish out a lot of damage and cripple limbs if they make it to you. Canyon Runner and Aurelius may put up a bit of a fight, but you should be alright if you just lay on the firepower. They won't have much backup.
Once you kill Canyon Runner, search his body for the keys to the slave pen. You can then open the door and disarm their collars to set the Weathers family free (or you can use your Explosives skill or your Speech skill to blow them up, if you're feeling evil).
Once you've cleared the HQ and the barracks, report back to Astor to receive a big XP boost and some caps.
For an irradiated ruin, Camp Searchlight has a large number of quests associated with it. There's an option to collect irradiated dog tags from NCR trooper ghouls, help some scavengers with their mission in the town and carry out a raid on Cottonwood Cove for the NCR.