The Skeleton dude is probably the easiest boss in the whole game. He only has ONE attack...the fire breath. The pattern of his breath is the SAME every single time. He does three sweeps with his breath, and repeats it.
1. From left to right, sweeps down, so jump to avoid it.
2. From right to left, sweeps down, so jump to avoid it.
3. From left to right, sweeps up, so duck to avoid it.
As for the part when you have to close the door on his head, yeah, you gotta mash the Y button, and FAST. You can easily do it with your default weapon (what you have after you die). For myself, I had to actually use my index finger to mash the Y button (thumb wouldn't move fast enough). But that's not a problem since you can start mashing AFTER you avoid the third sweep of fire.