As others said, you should disinfect anything that raw chicken touched with bleach. Even at ‘full strength’ (right out of the bottle) it’s not going to hurt your hands and it’ll be out of the air in a few minutes so no protective gear is required, just don’t get it on your clothes or anything else you don’t want, well, bleached.
You can also get any number of disinfecting sprays at your local grocery store/Target that contain bleach for cleaning counters or sinks. Spray it on, wipe it off and you’re good. Again, don’t get it on your clothes so pay attention to what you’re doing and use a paper towel that you can just toss in the garbage.
Personally, if I’m handling chicken, I have the dishwasher open with the rack pulled out so that I can just toss everything right in there without having to touch anything (like the rack or the outside of the washer) and then wash my hands or take off my gloves. Then spray down the counter and continue cooking knowing I don’t have to worry about cross contaminating anything.
ETA just for the record, full strength bleach won’t hurt your hands if you get it on them (and rinse it off) but it can potentially make you sick if you eat/drink it, so you do have to rinse that off of food prep areas if you’d use it on a cutting board and go right back to using it.