Enter the Gungeon - Resourceful Trophy/Achievement "Guide"
So, after years of waiting, players can finally fight the rat, but unlocking this fight is going to take some work.
To get to the rat, players need to repeatedly play the first two levels and slowly work towards buying the key. This is cumulative, so you can do 5, 10, 50 or even 500 and do the rest on a later run, with every run after you buy it costing around 120.
While you're unlocking the key, players also need to get notes. I don't know what triggers five to spawn, but I can say one through four can be unlocked normally and the sixth one is unique to mimics. I'm guessing you need to go to the lair and fail, since I got both immediately after fighting the rat, so I'm guessing that is it. For those who're like me and don't want to wait, you have a 1:16 chance of getting there without the last two.
Anyway, once you get the key and have at least four of the notes, you need to progress to the third stage and look for a room with rats in it. Once you find it, start looking for a hidden image on the floor and then run around it until a gate appears. This requires a single normal key to enter and will appear on the mini-map once you locate it, so try to do that before heading there.
Once you're inside, follow the cart to the end, use a blank to reveal a secret location, go to the end of there, use another blank and finally use the key to open the gate.
To get past the rats maze, you need to look at the notes and see which direction the cheese is pointing. Sadly, everyone has a different location, with 1024 different solutions, so odds are my path won't match yours. As for my last two, I actually just did the opposite of my first two and it just so happened to be the correct solution to my missing pages. As I mentioned above, two missing pages leaves 16 possibilities, making it fairly easy to guess it.
As for the boss, which I did horrible against, there are three phases. The first he doesn't have too much health, followed by a much harder stage and then a Punch-Out-esque fight. I could not do the fight, I honestly forgot I could block and thought I had to dodge, but winning it is not required for the trophy/achievement.
The best advice I can give is save your blanks for the second fight and try to learn the best way to counter. Typically the cheese attack has an opening, with the quick attack having somewhere safe to stand, making them pretty easy to overcome.
The robot follows a similar pattern, though I'd suggest just using a blank to get past any part you can't do.
After defeating him you'll get some loot, the chance at the Rats gear, the ability to snag the serpent for the advanced fight and a shortcut to stage five or the ability to also do stage four. Winning the fight gives the rat ring, which is useful in certain situations.