The new DLC has introduced easier methods for obtaining this achievement. All credit goes to AstroNightmares:
“I found a very easy way to get the achievement with the boss rush mode. With the new DLC there is an option to do a run with only rainbow chests, one per floor. Which has a much higher chance of having a clone in them than being lucky enough to find it on a normal run. In my second or third floor chest was a clone. I managed to not get hit by one of the bosses from the first four floors but of course got hit by the fifth floor boss. When I was revived by the clone on the first floor I killed the first boss without getting hit giving me a second master round for the first floor making a total of five, and the achievement popped.”
To earn this achievement, you must flawlessly beat 5 chamber
bosses. Optional bosses (Blobulord in the Oubliette, the Shadow Magician mini-boss, etc.) will not reward the player with a Master Round.
There are several methods to earning this achievement, each requiring a great amount of skill and luck.
1. The first and most obvious option is to flawless all 5 chamber bosses in one continuous run. Attempt this once you have invested several hours into the game and have learned boss patterns. Stock up on blanks and hope for S-tier gun drops.
2. The second, easier method is to unlock the Clone item by returning Ox's arm. Learn how to do this here:
The Clone item allows you to restart on the first floor upon death, keeping all weapons, items, and health upgrades. Hypothetically, you could earn several Master Rounds on your initial run and then earn your remaining Master Rounds on your Clone respawn. Just cross your fingers that Clone shows up on a strong run.
3. The final method requires the player to unlock Boss Rush. To do this, you must unlock all floor shortcuts (not an easy task). Learn how to do this here:http://enterthegungeon.gamepedia.com/Shortcuts
You have 3 chances to earn a Master Round in each of the first 4 chambers. However, you will still need to flawless the 5th chamber boss in one try.
The achievement will unlock as soon as you flawlessly beat your 5th chamber boss, before you even pick up your 5th Master Round.