The way you can complete this challenge is first, get good. If you so wish, after hours and hours of practice and runs, you get used to the challenges and the game becomes more manageable. The Floor 4 boss will have a special modifier, but if you're good at the game the modifier doesn't make them significantly more difficult. The Dragun will attack more rigorously and you can't take two hits in succession, but if you've beat the game a good amount of times it's feasible to complete it on just a normal run.
If you want to cheese it, the Gunslinger makes this challenge significantly easier (though unlocking him is very hard), and you can also use a Rainbow Run to make this even more easy. Just repeatedly play until you get a boss destroying gun, and breeze from floor to floor collecting A and S tier items, evaporating the competition. The Robot is a good pair for a Rainbow Run, as the junk from the normal chests will upgrade his damage output tremendously over the course of the run, and after a few attempts I was able to complete it using the Makeshift Cannon and Gunther.
If you haven't yet unlocked most of this game's really good items and weapons, you should probably try to target some other challenges before attempting this, as the items you can gain by doing so are immensely helpful. Also remember that the Enter the Gungeon wiki is extremely helpful for understanding weapons and items, and that you can use the wiki as a reference for many aspects of the challenge run, how different items and guns behave, and how to unlock them.