Have you tried hypo-allergenic food? Some cats are allergic to ingredients
which are added to most ordinary cat food (wheat, cheese & milk are common
ones, and also some additives). You can buy cat food which doesn't include
these things specially made for the allergic cat. It's more expensive but
may be worth just trying to see whether he still vomits after eating it or
not.

How long have you had Charlie? Does he have previous owners you can contact
and ask about this?

Is he losing weight? If not then he must be keeping enough down to maintain
himself.

About 2 years ago my cat quite suddenly started vomiting after eating cat
biscuits. He'd eaten these without a problem for 2 years before that (since
I had him). I tried different brands and flavours but it made no difference.
He was fine with his wet food though. I mentioned it to the vet who advised
me to try him on the hypo-allergenic type. Hey presto, no more vomiting.
I've used the hypo-allergenic dry food ever since and he's never vomited
after eating since. (He still happily eats the wet food without a problem.)

Have you tried something like raw chicken wings (never use cooked ones)?