In a sense, the major characteristic is that there was no such thing as "the" ancient Greek law code. The area that is now modern Greece in antiquity consisted of independent city-states inhabited by people who spoke Greek. Greece was not unified until it was conquered by Macedonia.
Each Greek city had its own code of laws and there were major differences in legal and political systems among the city states.
There were a few commonalities among the laws and societies of these city-states. Slavery was ubiquitous and slaves were legally treated as property, not as people. Greek city-states were patriarchal, with women subordinated first to their fathers and then to their husbands. Women, slaves, and children were all part of a male dominated household. Women generally could not represent themselves in legal cases but male guardians acted on their behalf.
The legal statutes of most city-states incorporated a concept of private property and had penalties for theft or destruction of property. Murder was generally illegal, except under certain circumstances (self-defense, a man finding someone in bed with his wife). Penalties for most crimes included execution, exile, sale into slavery, or fines.