I inherited a cat a year back that was doing this exact thing. The vet said the pancreas was failing and that was causing him to not be able to digest food. She said it was probably caused by him not eating good food.

Get your cat more expensive cat food, one without the first ingredient being ground corn, or gluten meal, or meat-byproducts.

You need to get cat food that is usually a bit more expensive but it is worth it. The first ingredients will be whole white chicken, or just salmon, or beef, and not fillers.

Your cat is probably becoming intolerant to the crap that is in the shit food they make for animals these days and they purposely jack up the price of the good food just to make you buy the cheap shit, trust I did it for a while and lots of people still do just because it can get really expensive if you have lots of cats. Just one cat it won't be too bad, you will just have to tough it out for the good of the cat.

Also get him some soft food and some hard food and feed him just one for 2 days and see if he pukes and then try the other and see if he pukes and it will let you know if one or the other is triggering it. My cat still to this day will puke if I let him eat to much hard food, he can only snack on it little bits at a time.

Also you need to feed your cat periodically every 3 - 4 hours and half the amount you use to give and then work it up but to get him back on track feed him many times throughout the day, less quantity and it will help him digest better.

He is throwing up without eating now.... that is because his stomach is empty from puking all the time. You need to correct this fast because the stomach acid is eating away at his lining at it is causing him to puke up bile because he has no food in him and no nutrients.

Hope all this helps...honestly it is all the research and things I tried with my cat and he was horrible when I got him, constantly puking and was so thin.