I can't use bleach either, my throat swells up for 2 days, my eyes turn red, and I get a splitting headache. I understand you
Vinegar has many many applications, like Traveler said.
I also use soda cristals in very hot water (but wear gloves to protect your hands) with some ecological dish washing detergent. It is the best degreaser, it does what all other products make you believe they do.
And for general cleaning I use natural soap with linseed oil or black soap. Marseille soap is also very good, and it smells great.
For doing the dishes, I use ecological detergent, it is really gentle on the skin, I do the dishes without gloves, so that's a good sign.
I was always sensitive to chemicals, but since starting treatment, it has become much worse.