How can I --a civilian -- find out if someone I know has a criminal
record in any county in the state of California?

Hi
Some of the options to consider:

1. Online research, which you will have to pay for:

://www.google.com/search?sourceid=navclient&ie=UTF-8&oe=UTF-8&q=criminal+record+research

2. Use free online services, which are not always reliable or up to date:
http://www.journalismnet.com/people/uscrime.htm#tools

3. Contacting courthouses by physically, by phone or written request.

A good article can be found here:
http://records.4mg.com/criminal.htm
"There is no such thing as a national criminal records check.  There
is one "nationwide" criminal database, the FBI database...The FBI
database is NOT public record and cannot be legally accessed by anyone
other than criminal justice agencies."