I'd say your best lifeline for getting this achievement is to have a slew of Stealth Boys on hand. Even with relatively low Sneak skill, a Stealth Boy is invaluable to have towards racking up your pickpocketing total. Stealth Boys can be hard to find in the Mojave Wasteland, but this entry on the Fallout Wiki should help tons: http://fallout.wikia.com/wiki/Stealth_Boy_(Fallout:_New_Vega...
After acquiring Stealth Boys, you need to get to some populated areas in order to rack up as many pickpockets as you can before the Stealth Boy wears off. Best bets are to go to the casinos on the New Vegas strip, or to other places such as the Great Khan Encampment or towns such as Novac or Primm.
Last word of advice is to have Lily as a companion, for the "Stealth Girl" perk doubles the duration of Stealth Boys.