Just clean it with a bit of dish detergent and you'll be fine. Squirt a bit of detergent inside and fill with warm water from the faucet. Shake it around for a bit and then let it sit for a few minutes and rinse it out. If you have a brush or something you can scrub it out if you like. Scrub the mouthpiece with tooth brush and dish detergent or something and it'll be fine.

Assuming tap water was left inside the bladder, it would have been chlorinated to prevent stuff from growing in it. In a sealed system the chlorine should not have evaporated out. The plastic used for the bladder also has something inside that prevents growth of bacteria and such; this does not imply it is purification device.

For sanitizing things, the recommended dose is 1 tsp per 1 gal of water according to CDC so if you want to use bleach, you won't need to exceed this. Rinse it out well afterwards.

Not knowing what is in your dish detergent, I do not recommend mixing bleach with it.

Honestly, I think the bigger hurdle it sounds like is getting over the ick factor that someone drank out of the thing before you. There is bacteria everywhere including flora that is naturally present in your skin and gut. Most of this stuff won't harm you.