how to extract copper using living organisms?

Not easy. Very few organisms metabolize copper.
(Think the horseshoe crab is one).

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Here's the REAL deal:

The copper minerals are usually sulfides, such as CuFeS2. Bioleaching uses living organisms to extract metals from their ores. Two bacteria, Thiobacillus ferrooxan and Thiobacillus thiooxan, catalyzes the oxidation of these sulfide minerals of copper. Sulfide minerals are insoluble in water or acid solutions unless they are first oxidized. The overall equations are shown below:

1. 4CuFeS2 + 11O2 + 6H2O Thiobacillus ferrooxin >> 4CuSO4 + 4Fe(OH)3 + 4S (1)

2. 2S + 3O2 + 2H2O Thiobacillus thiooxan >> 2H2SO4 (2)

Note that the copper in produced in the first reaction can now be collected in a solution!
This process is more efficient at higher temperatures ( as the reaction proceeds, passivating layers build up that prevent further oxidation, heat destabilizes the layers) and efforts are now being made to find a optimizing bacteria.

I've summarized some key points to how this is done. The link below explains it in detail, it also provides a list of companies that bioleaches copper that you can look up for further information :) The link explains the process currently being used in those companies to extract the copper: it requires special equipment and preparation, so it's probably not feasible to do it yourself.

It's been done, but whether it's practical or not depends on what you're after. I wouldn't say it's useful for mining, but it's been proposed as a means of cleaning up metal contamination of areas near mines.

The idea is that you find a variety of plant (usually grass) that's resistant to whatever contaminant you're looking to get rid of, and yet accumulates it while growing. In this case, copper. Plant the grass seeds, and whenever you mow the 'lawn,' you collect up the clippings or harvest the plants and dispose of them as toxic waste. Whatever amount of copper that the plant accumulates in its biomass won't be in the soil anymore. The government of Canada has funded research around Manitoulin Island on this in the past. I don't know if they still are or not.

I suppose if you wanted to, you could mash up the grass clippings and extract out the copper chemically, but it wouldn't be very practical. Traditional mining methods would probably be faster and cheaper. The percentage copper in the biomass wouldn't be much, so you'd need to collect tonnes just to get a few grams, if that. Also, it's also quite possible that the process you'd need to extract out the copper would involve chemicals that are every bit as toxic as traditional mining, leading to another set of polluting problems.