On a recent vacation in Hawaii, I visited a jewelry shop in Waimea.
There was a ring with a 3-carat black opal, turquoise and blue color
(no reds).  The owner told me that the pattern was "chinese character"
which I understand is very rare. The stone is all opal, no backing,
thick (high domed)and oval shaped and is very clear, no sand or
imperfections. I was quoted a price of $6,250.00, and the owner
claimed the ring would appraise at $15,000.00. I am considering
purchasing this ring, but I don't know enough about opals to know if
the price quoted is reasonable or if it is overinflated. My question
is simply: Should I purchase this ring, or keep looking for a better
deal?

Hmm, I would be suspicious of a jewelry store that claims that it is
selling something at less than half an appraised price.
Here is a more or less wholesale seller's prices for larger black opals:
http://www.kashmirblue.com/Coloredstones/kbopalblack.html

Here is an informative site about opals, relative quality and value, but no prices:
http://www.wired-artist-jewelry.com/OPAL_VALUES.html

But here is one with prices:
http://www.gemworld.com/black.asp

I won't bother you with eBay, but there are a lot of black opals on
sale, and not for 4 digit prices.

This site is also very informative, and does suggest higher values:
http://www.anoraanna.com/pricing%20guide.htm

And this one also suggests what you should ask a jeweler:
http://www.opal.asn.au/information.htm

Armed with this information, maybe you will be more ready to ask what
"Chinese character" means.

My personal opinion is that you did the right thing by not buying
immediately, recognizing your lack of knowledge before making such an
investment.

Before making a decision, take a look at this over-37 carat unset
opal, and at other opals offered on the same website - 
http://www.opalauctions.com/item.php?id=18076