I would like to apply for citizenship in Greece, I am an American citizen, and would like to obtain dual citizenship

Hmm

It might be possible , though you would have to look into it as many countries don't allow Dual nationality.

For Example : A child born in a foreign country to U.S. national parents may be both a U.S. national and a national of the country of birth.

Or

A U.S. national may acquire foreign nationality by marriage.

Update

If this website statement is true then yes you can ( American and Greek dual citizenship (http://livingingreece.gr/2007/06/03/dual-citizenship-american-and-greek/) )

Hope this helps.

Not easily. See the link above.