You must be level 10 and have spoken to Lester in order to have bounties put on your head. This can be done solo, with randoms or in a friends/crew only game.
Solo:
Just keep stealing cars and eventually one of the NPC characters will put a bounty on your head. Just survive 24 hours (48 in-game minutes) and the achievement will unlock.
Friends/Crew:
The person setting the bounty must also be level 10 and have spoken to Lester. Get them to set the bounty and then survive the 24 hour time period.
Randoms:
Eventually someone will put a bounty on your head, especially if you piss enough people off
Surviving the 24 hour period will be slightly tougher since other randoms will want to cash in so here are a couple of tips:
- If someone places a bounty on you but leaves before the 24 hours are up, it will still unlock.
- Get in a helicopter or plane and fly for the entire time. Pick a very tall building to land on if necessary.
- Get a fast boat (Sea Shark) and cruise around the waters for 24 hours.
A couple of additional notes:
- Getting shot by the police doesn't seem to ruin the achievement.
- Time spent in Los Santos Customs does NOT count.