How can I use bitter or hard water for cooking and specially in soaking beans?
In most countries, hard water is caused by Calcium and Magnesium Carbonates.

These can be easily removed by using an ion-exchange apparatus (best results) or can be converted by adding an acid to the water.  In the case of vinegar:


  2 CH3COOH + CaCo3 → Ca(CH3COO)2 + CO2 + H2O


So the Calcium is not gone: it's just converted to Calcium Acetate but that is much more soluble in water than carbonate, so it'll seem softer.

The same is true for MgCO3.

How much vinegar and how long does this take?

Overnight should do the trick unless you live on one of the poles.  ;-)
How much??? I cannot tell you over the Internet so experiment: 


Start with 20 drops per Litre
Use less if that's satisfactory until it becomes unsatisfactory.
then go back up one drop and that amount will be the best volume for your water...