The counterpart to New Vegas Samurai, the 10,000 damage for this achievement must come from using your bare fists or Fist/Glove weapons. A respectable strength (5 or higher) is required for this achievement, and a high agility and stealth are also recommended. To expedite this achievement, try to make a character that focuses on getting critical hits through stealth and VATS. That will greatly increase your damage output and effectiveness.
For some of the Damage achievements, you can collect as much ammo/weapons as you can and make a save on the Strip. Here, you can kill/destroy all citizens until you unlock the achievement.