EDIT 30 Nov 2020: The meta in this game has changed a lot since I wrote this solution. I am no longer playing the game so am not well equipped to say whether these tips are still good ones. Feel free to add comments with your own ideas and I can incorporate them into the solution, or otherwise please take this solution with a grain of salt.
When going for this achievement, keep a few things in mind:
1. You need to stay alive long enough to make it to the end, which is when the crown spawns. I recommend avoiding encounters you aren't certain you'll win, and grabbing the hacks you feel most comfortable evading with (for me, that's Slam and Teleport - more on this later).
2. You'll want highly or fully fused weapons to kill the crownbearer quickly.
3. If you have good squadmates, it's likely easiest to make it to endgame in squads mode. I generally found Solos easier to reach the crown round in, but that's for you to decide.
All said, here's my recommended kit:
- Skybreaker, fully fused (this gun is amazing, huge splash damage and kills in 2 shots when fully fused. Aim it at the floor or wall next to the crownbearer to kill them extremely fast)
- Riot one or Ripper, fully fused - Riot one is my preference because it does huge damage. The Mammoth shotgun can be a good choice if the crown spawns in an indoor, close quarters area, but that's pretty rare.
- Slam + Teleport, fully fused: These provide the mobility you need to (a) stay alive until the crown spawns, (b) keep up with the crownbearer when they're running away, and (c) get vertical so you can shoot the Skybreaker down on the crownbearer.
To state the obvious, you should never pick up the crown when going for this. This is another reason why Solos is easier than squads unless you're playing with teammates who know what you're trying to do.