This achievement requires you to collect 4 billion studs, which isn't as daunting as it appears. You will want to get all stud multiplier bricks as fast as you can, the locations of which are as follows:
Studs x2 - Mission 2: Trouble in Stir
Studs x4 - Mission 5: Dirty Work
Studs x6 - Mission 12: High Steal
Studs x8 - Mission 13: Disruptive Behavior
Studs x10 - Mission 15: Fly Me to the Moon
Unlocking all of these will grant a x3840 multiplier. I was able to get the x4 and x6 in story mode, but the others you'll want to grab as soon as you start your freeplay versions of the missions. Grabbing even just one of these will make achieving Lego City Hero in each level a breeze.
Otherwise, just go through completing the game without worrying about this one. If you're still short by a billion or two by the end as I was, there's an opportunity with the completion Stud Fountain. Once you've gotten the last collectible, a stud fountain will appear and you'll be teleported to it. Activate all the multiplier blocks and run around the fountain for a while, as it pours out blue studs, and the achievement will unlock soon after.