Reduce the AP Cost of Unarmed Attacks in VATS.”— In-game description
To complete this challenge, the player must inflict 10,000 damage with unarmed weapons.
Beautiful Beatdown reduces the Action Points cost for unarmed attacks by 10%.
In order to get this perk, you first have to complete "A Fistful of Hollars," which will unlock "Destroy Something Not So Beautiful," which in turns rewards you with Beautiful Beatdown. The "Old-Tyme Brawler" challenge has the same requirements, but will only give you the achievement, not the perk.