Bleach is unnecessary. The main defense most viruses and bacteria have is a lipid membrane that breaks down easily with soap and hot water. Ordinary Dawn detergent and hot water will do.
Recently I've tried a few other liquid dish soaps, though, and found Palmolive Ultra Antibacterial may be even better -- it gets dishes and my hands even more squeaky clean and free of oil or grease than Dawn, which is already very good. It adds lactic acid as an antiseptic, which may be useful on our hands but probably unnecessary for our bikes.
If you want to try heat as a disinfectant, an ordinary steamer wand for de-wrinkling clothing works fine. I use it for all my fabric covered furniture too once or twice a year. It emits steam with very little force so there's little risk of blowing cooties around, unlike hair dryers, hot-air paint strippers and other stuff I've seen suggested.
Or a heavy duty steam iron, but those are really awkward and heavy. A clothing steamer wand is lightweight and easy to handle.