Hmm... I am not sure about bleach but I think the residual will be as harmful to your fishes. Here is what I will do. Pour all the water away. Clean the tank with clean water. The most I would use might be some mild dishwasing detergent. Be sure to rinse it out properly. Spread the gravel onto a tarp and wash with clean water too. Let it dry really well under the sun. Set your tank back up with the gravel with new water after. Don't put fishes in there yet so tap water will be fine.
I would buy or use a filtration system where I can get a used filter of the same size from someone else - friends or local fish store that is from a healthy clean tank. Use that filter on your filtration system. Why use this? A used filter will contain all the proper bacteria that is needed to jumpstart your tank. Quicker than waiting for it to develop naturally. Eg. break down ammonia, etc.
When I start a new tank, the water is always not clear but don't worry about this. Let it sit for a min of 3 to 4 weeks. Change the water like normal. You can look up the frequency to change water based on size of tank. I do a water change every fortnight on my 29 gallon tank.
As the water clears up, I would start with some hardy fish (fish that do not die easily) like 4 or 5 Albino Corydoras and just let them be in the tank for another 2 to 3 weeks before adding other fishes.
Here is what I have used:
- 2 liter coke bottle
- tubings that is smaller than the coke bottle cap (~5mm)
- airstone
- super glue
- knife
- yeast
I cut a hole that is just a tad smaller than the tubing diameter on the center of the coke bottle cap. Stick the tubing through and apply superglue around the hole to seal it properly. Attach the airstone to the other end which you will set to the bottom of your tank.
You can do without the airstone if you don't have one. I just stick my tubing under a decorative stone to hold it down.
Mix half a cup of yeast with warm water. Stir to mix it evenly. Let it sit for about a minute or two and then pour it into your coke bottle. Add more water to fill the coke bottle about slightly below 2/3 full. The yeast should start producing CO2 in a day or so. The measurement above are subject to change. Experiment with it. I did it a few times before I found the right amount of yeast and water. Quality of yeast will also affect the production of CO2 so do a trial and error until you find the correct amount of what you need.
I am sure there are better methods out there but I find this to be very cost effective. You get to enjoy your coca cola or favorite beverage before using it!

Most aquariums owners I know already have tubings. If not, just head to your local fish store to buy some cheap ones. The wrong mixture combination will produce too much or too little CO2 so find the right mixture for yourself.
Just a word of caution. If you use an adjustable valve to control the release of CO2, be aware that you may have an exploding bottle on your hand if it produces too much CO2!! Do at your own risk!

Good luck.
RB
Wah... detailed! Wuuu... thanks... Corydoras are hardy? Didn't know...