This achievement unlocks naturally by progressing through the main story and should be unmissable. In case you are having trouble unlocking it or completing the mission here is an explanation.
After the mission "Russian Revolution", the next mission is mission #25 "Roman's sorrow". You will be told to call Roman. when he answers it sounds like he's in trouble and you need to get to him as fast as you can. Once you make the call the mission will start so head to the alley where he's hiding out by using your GPS to guide you.
Once you get there a short in game cutscene will play. Roman asks to go back to the apartment and pick up your stuff, so make your way back to your apartment.
Roman now asks to go to the taxi depot.
Roman says that it's time to head to Bohan where you can stay with Mallorie's cousin.
*** Spoiler - click to reveal ***As you go, Roman will say how much he hates you and blames you for all of the bad things that have happened to him since you arrived in Liberty City. He'll also talk about the dream he had, and how he was planning to marry Mallorie, but the ring was lost in the fire.
Make your way to the new safe house now.
Once you arrive at the new safe house, you'll see a short cutscene.
*** Spoiler - click to reveal ***After this mission, you will no longer have a safe house in Broker, but do have a new one in Bohan. You can still save your vehicles at the Broker parking spaces though.
Brucie will also text you and let you know that you are now able to do some Vigilante missions from the cop computers.
The achievement/trophy should unlock shortly after the cutscene with Mallorie ends.