Just played through this. I like the presentation a lot, both the art and music were definitely very enjoyable. I think an alternate control scheme would probably benefit the game a lot though, as jumping with W was pretty awkward.
Some other nitpicks:
The jump sound plays whenever you press W, even if you're in midair, which lets you create a cacophony of jump sounds pretty easily. Kinda funny, but you might wanna fix that.
Jump detection felt a bit finicky overall, especially when jumping near edges.
The bosses were pretty fun, but it felt like they had too much health, probably because you got no real feedback as to your progress until you kill them. Maybe add some visual cues on them once you get past 50% health or something, or have them speed up their bullet patterns a little as you hurt them, so you know that you're doing something.
Also, I have to agree that the checkpoints didn't make a lot of sense, and neither did going back to the title screen every time you die.
Also, on the last two levels the game's music cut out entirely, except when the window wasn't in focus, and the framerate slowed to a crawl. Not entirely sure why, since it didn't look like anything particularly intensive was happening.
Besides that though, it was quite an enjoyable game. Keep up the good work!