Firstly, turn autosave off and make a back up save as you need to play honourably to unlock missions for
Lending a Hand.
There are five states but you only need four for this achievement, according to comments below, so thanks for the feedback:
1. Lemoyne (do this last)
2. New Hanover
3. Ambarino
4. West Elizabeth
For New Austin, you automatically have $250 bounty here by default due to events in Blackwater mentioned in the prologue and chapter one. It is possible to remove this bounty later in the game and comments say they did not/cannot get another bounty in New Austin, so don't worry about that state. To get a bounty, head into a town and cause chaos by shooting people and animals. It doesn't matter if you die when you are obtaining a bounty, the time only matters when you have all 5 bounties active.
Ambarino can be a bit tricky as there is not a major town so kill someone on your way there, mount them to your horse and then when you see another civilian, they will alert the authorities.
EDIT: Maxumulli0n has suggested an alternative way to get a bounty in Ambarino:
For Ambarino, stand near the train tracks heading east to west and stop the train in Ambarino. Eventually, the sheriff's will get alerted and take out as many as you can to get your bounty up.
There are several comments below for Ambarino alternatives as it can be quite tricky. Thanks to everyone who made suggestions.The last one is Lemoyne and I recommend going to Rhodes for this. Rhodes has a mill at the north-western part of town. Head there and to the room inside, going up the ramp. Use the sawn-off shotgun on the authorities that enter from the ramp or climbing up the ledge to the left. Once you have at least $250, let them kill you.
Once you respawn, head to the saloon and rent the room. Sleep 18 hours twice (max allowed) and then go upstairs and play blackjack for about 5 minutes. Head back downstairs and rent the room again, sleep for 18 hours and it should unlock. If not, sleep again.
You can not be attacked by bounty hunters whilst playing a mini-game, and they never started a pursuit whilst I was in the saloon. I stayed there for another 4 days after the achievement unlocked and nobody came.