Explain how archimedes might have used the concept of density to determine whether the king's crown was pure gold. (density of gold is known to be 19.32 g/cm3 )

In theory this should work.

If you take any shape of gold and put it in a container of water (the water being filled right to the top) and measure the overflow of water you get the volume of the object.

If you then measure its mass, you can get the density with this formula
Density = mass of object made of gold / the amount of water overflow 

Given the crude methods Archimedes had at his disposal, any answer between 15 and 20 for the density would have been acceptable. If the object has any other impurities in it, the density would fall well below 19 or even 15 if there were enough impurities in it.