To get a crown you'll need to access the crown vault, to get there you need to activate and fill the proto statue, then defeat the final boss while having a crown equipped.
The proto statue is a greenish-gray rhombus shaped device.
The proto statue spawns in the scrapyard and the frozen city (3-2 and 5-2 respectively.)
Once you locate the statue shoot it and it will activate. (As it activates IDPD units will spawn nearby.) Any enemies you kill near the statue will have their radiation sucked into the statue and slowly charge it to completion. Be careful not to damage the proto statue too much before it is fully charged as it can accidentally be destroyed this way, dropping any radiation it collected before it was prematurely destroyed.
(More IDPD units will spawn once the statue is fully charged.)
The statue is fully charged when the normally vacant center has green electricity and a green orb inside it. Once this happens, simply shoot the statue until a green portal spawns and enter it.
Then, simply find the crown pedestal and pick a crown. Once you have a crown, IDPD units will spawn as soon as you enter the next area. (So, 3-3 or 5-3 depending on the statue you took.)
The final part of getting this achievement is carrying your crown all the way to the final area, defeating the boss and sitting on the throne afterwards, unlocking the crown on that character allowing you to start runs with it on. (When you start runs with a crown, a single IDPD portal will immediately spawn in 1-1.)
Good luck!
Side note, when you are trying to access the crown vault you should attempt to prioritize killing IDPD units as they are deadlier than your average unit in the scrapyard and can be a big problem in the frozen city when combined with the regular enemies that spawn. The units that spawn after the statue is fully charged can be ignored as long as you can destroy and enter the portal fast enough.
The IDPD units come in 3 variations, a smaller, mobile unit that shoots in bursts and has a dash (The grunt.), a medium sized unit that pulls you in with telekinesis and shoots you with an IDPD colored slugger (The inspector), and a big one that sprays tons of bullets and has a crystal-style bullet shield that can reflect your bullet projectiles back at you (The Shielder.)
My tips for these are as follows: Grunts have relatively low health, and can be rushed down easily shortly after spawning. The inspector is kinda the same, be cautious as when they start pulling you in with telekinesis as it means they're ready to shoot, you can try to avoid his slugger projectile by strafing horizontally relative to the inspector. The shielder is very hard to rush down due to his higher health pool and shield ability, but you can still get free damage on him as he spawns. While he's shielding he's immune to all attacks except energy and melee attacks, so try not to shoot at him during his shield except with energy or melee attacks.