If you're looking to get this out of the way like I was, the fastest way by far is just to make a save, then shoot up 3 different areas that house 3 different factions. I would suggest:
Spacer's Choice - Town of Edgewater on Terra 2. This will be one of the first busy areas you frequent in the game.
Groundbreaker - Anywhere on the ship. You can travel here shortly after getting your power regulator in the 'Comes Now the Power' quest.
Auntie Cleo - Town of Roseway on Terra 2. Talk to Gladys on the Groundbreaker to get a quest about a distress signal. Talk to ADA back on your ship to unlock a new area to travel to on Terra 2.
The achievement seemed to unlock around 85% for me for my third faction, but I got the first two to 100% anyway as I was grinding for kills at the same time. To check your standings with the factions, it's under your Character Tab under Reputation.
Again, don't forget to make a save if you choose to do it this way :)