I haven't noticed any problems with bleach or rinsing it afterwards.
Everybody is always scared they're going to introduce off flavors with chlorine, but you fix that with rinsing. Then they're afraid they're going to introduce bacteria with the rinse water, but that's not an issue if your water is treated. I've had neither issue, however YMMV.
Depending on your brand of dishwasher soap, you might have just bleached your bottles anyway albeit probably not in enough of a concentration to sanitize your bottles, but if your water was hot enough you may be okay. In that case, I hope you rinsed it well.
The other type of dishwasher soap is enzymatic soap which doesn't do anything to the bacteria already there. It may or may not affect the outcome of the beer. I don't have a firm grasp on what those enzymes target.