Remove all gold colored parts from your PC. Take off the computer casing and take all the parts out. Look closely at each part and set aside any with gold colored parts. Edge fingers, processors, chips, pins circuits and other parts that are gold colored are what you want to get. Sort out any brass which will be more yellow than gold and separate that from the gold.
Send all gold colored parts to a refinery. It will then give you an offer for your PC gold scrap which you can accept or decline. If you accept, the company will send you your money. You can send whole parts in but you might get more for separating all the gold. That is very time consuming though so keep that in mind. Be sure to remove as much metal, plastic and other junk as you can to maximize your profits.
Recycle all other metal scraps at a scrap metal recycler such as American Metal Recycling. Most scrap metal recyclers accept scrap metals but some specialize in certain metals. Make sure the recycler you choose accepts non-ferrous metals before taking them in.