The biggest tip I can give for this is: buy and use the Neptune Gun upgrade! It makes all the difference. But be careful: when you get hit by an arrow from the Pyro Archer (Flameslinger) you get stunned for a moment. So if you're still pressing A you will fire an unwanted Harpoon.
When that happens, just die. The game will say Gill Grunt needs to rest, but just press Y and you start right from the beginning of the boss fight. No need to replay the entire level.
And try not to eat the health pickups when your at (almost) full health. That's just a waste. Save it for when you really need it.
I also recommend wearing a hat that gives you some armor. Gill Grunt's basic armor stat is laughable.
Last tip: remember the patterns for the fire spell. When you get caught in it you lose A LOT of health.
Maybe not the best solution out there, but, yeah. It's better than nothing I guess. It might help someone until someone writes up a better solution?