EDIT: As of 01/12/17, the number has changed from 100k to 35k. However, most of the strategies still remain the same.
This is a bit of a bastard. And before you ask, no, you don't have to hoard it all in one go, it is cumulative.
Most enemies will drop about $10 when you blow 'em up. The ones that drop heat seekers after they die will drop $20, and the bosses drop $1000 sometimes. I say sometimes because, for some reason, level 20 and 30 didn't drop their cash pickups, and it might be do to the fact that the cash disappears faster than you can blow 'em up. If you are going for this, you're gunna want to stock on health, since you'll be running into a lot of enemies and gunfire, thanks to there being no magnet power-up or something.
As it stands, even doing that strategy put me nowhere near the achievement. I got as much as I could by level 40, and after the boss, I unlocked the achievement "Rainy Day", which is for grabbing $50000. The game doesn't continue after level 40 either so, y'know.