The problem is that bookmarks do not know about TeX fonts. The Greek in the question is very likely based on font encoding, using Greek characters instead of the standard Latin alphabet and ligatures for generating accents. These are features of TeX's stomach. Bookmark strings, however, are simple text strings. Mainly only TeX expansion (TeX's mouth) is available for processing bookmark strings. Example, if the code for processing bookmarks in hyperref gets the character C, it does not know which font is used in TeX to typeset this glyph, neither it does know the language. Based on the features of font encoding and ligatures a string might contain many C, but meaning different glyphs.
Ways to get Greek bookmarks
Bookmarks strings can be encoded in PDFDocEncoding or Unicode. For Greek Unicode is needed (regardless of input encoding) in all cases:
\usepackage[unicode]{hyperref}% or pdfencoding=auto
Package hyperref's \pdfstringdef directly supports UTF-8 of utf8.def or utf8x.def. If UTF-8 is not an option for all the Greek text, then it can be limited
to bookmarks:
\section{\texorpdfstring{<Greek text>}{<UTF-8 for bookmarks>}}
LuaTeX or XeTeX have native support for UTF-8/Unicode. These big chars (with character codes greater than 127) are also recognized and supported by hyperref's \pdfstringdef.
If markers for the Greek stuff can be introduced inside the bookmark strings:
\section{\texorpdfstring{...}{...<BeginGreek>...<EndGreek>...}}
then it would be possible to catch the partial Greek strings and transform them
to a form that is supported for bookmarks (via an expandable implementation or using the post-processing hook). But it quickly becomes nasty, if the Greek stuff is nested or includes Latin fragments (via \ref, …). Also the whole translation process from Greek in Latin form to real Greek has to be implemented.
Short Greek stuff can also be set by using \unichar with the Unicode number
in the second argument of \texorpdfstring. But for long texts this method is not very convenient.