Crowns are earned by completing Co-op missions and Special Operations. You can also earn them from daily streak bonuses, or by successfully completing certain 3-hour contracts in Versus if crowns are the payout.
Once you complete Co-op or Special Ops missions, you'll be awarded with crowns. The more kills your squad gets the more crowns you'll earn and the faster you complete the mission the more crowns you'll earn.
The missions change every day, so if you max out on crowns today, come back tomorrow.
Once you've earned 100 crowns cumulatively, the achievement will pop at the end of the match.
You can check how many crowns you have by looking at the bottom of the screen in any lobby. From the main menu, just press "Play" and they will be displayed on the bottom.