This guide is more about how to spawn an Advanced Boss and not tips on how to beat it.
So there are only two ways I'm aware of, as of writing this, that you can get an advanced boss to spawn:
1) You defeat the Resourceful Rat and find the serpent in his loot room after the fight.
2) You have the item Weird Egg and you drop it onto fire to hatch the serpent from the egg.
Defeating the Resourceful Rat requires you to obtain the Gnawed Key from the second floor, finding the trap door to his area on the third floor, and beating the three phases of his fight.
The first phase is quite easy and you shouldn't struggle much unless you have a bad setup. The second phase is also quite easy with some practice. However, the third phase is where it really matters. It's a Punch Out mini game and you need him to drop two keys from this fight.
If you jab him immediately then he'll drop a key. If you counter his attack when he glows with a jab, then you'll build your super. You will lose your super attack if you get hit before you use it. You need to kill his first phase, or any phase, with your super attack for another guaranteed key.
After the Punch Out mini game, you'll collect your loot and drop down to a floor with some pedestals and chests. You'll want to use a blank here to open up a secret room that will contain the serpent. You need use your two keys here to access the serpent. To awaken him, you'll need to drop four items/weapons in front of him. After this, he'll follow you for the rest of your run.
Once all of this has been completed, you need to make it to the Dragun. It should be a fairly easy feat if you've beaten the rat. Fight the Dragun as normal until he goes into his second phase. The serpent will be firing at you the whole time. Once he goes into his second phase, the serpent will combine with the Dragun and the Advanced Dragun will form. This is a harder version of the fight with more bullets to dodge, new attacks, and he has also healed to full health. It's a little tough, but doable if you keep at it. Try to build up as much health as possible and use your blanks whenever needed.
Once you defeat the Advanced Dragun, the achievement will pop.
If you find the Weird Egg on a run then you just need to drop it onto some fire by holding up on the d-pad. This will hatch the egg. After that, you just have to make it to the Dragun and beat the Advanced version.