Daniel, I did this as a hobby for a few years. I was in the computer industry and I had access to tonnes of old computers. I will go through it in a minute, but firstly I gotta say that it is massively time consuming and very toxic. And you need lots of material and the older the computers the better... and I mean Pentium 1's. The best chip was the pentiumpro which came straight after 486's. Better than this is the boards and chips out of mini's and mainframes from the seventies and eighties.
The boards and components were gold plated and the chips had fine gold wire in the processors. You snip and break down the components and take off what is not gold (as much as you can). Then you put the material into pool acid (hydrochloric) in proper laboratory glassware. Leave it for days. Stir it, warm it, be careful with it and don't breathe near it. You'd really need a fume cabinet.
Crush the processors and put them in pool acid.
After a while the gold plating would float off after the base metal was dissolved from under it. Repeatedly, you wash the mash and filter it. Clean the mash out of the filter paper (could be fat filter paper from burgerbars)... you need funnels and retort stands, filter paper, beakers, flasks all of that stuff.
Keep on soaking it in HCl until you feel that most of the base metals have be dissolved in the acid. Then wash the mash again and again then you mix up a deadly acid called Aqua Regia or Kings Water. This stuff dissolves anything including gold and lungs. Aqua Regia is one part nitric acid and two parts hydrochloric acid. Extreme professional care must be taken. Pour the aqua regia onto the material and the gold will dissolve and create a solution which is auric chloride. You will have a beaker full of what looks like golden piss.... beautiful. The gold is in this solution and to precipitate it out you can use sodium metabisulphide, which you can buy from homebrew shops. It is used to clean the beer lines in pubs. You wouldn't want to breathe this stuff, either. You put a couple of tablespoons of this powder into a beaker of hot water and it dissolves.
Then pour the sodium metabisulphide solution into the beaker of auric chloride and amazingly the gold becomes a powder and the golden solution becomes dark brown and the gold falls to the bottom of the beaker as a brown powder which looks like milo or cocoa powder. The solution becomes clear as the gold falls out.
Then pour off the liquid and wash and filter the powder. In the end you have this brown powder which you mix with a borax flux in a melting dish. You take care to heat this dish with an acetylene torch (map gas bottle is good). At this point you could blow the powder into the air.... careful. The powder melts into a button of very pure gold. lovely
Don't do it. It is very time consuming, the chemicals are dangerous and expensive, you have issues with disposing of toxic waste, you need masses of old IT equipment.
This waste is sent over to china, india, phills and other Asian countries where they don't have the same laws as we do. That means that their kids in the nearby schools are breathing this poison.
Best leave it alone. accept that it is not a viable thing to do here.. I recovered about four ounces in three years.