If you are a U.S. citizen temporarily residing outside of the United States, or a member of the Uniformed Services or one of their dependents, you are eligible to vote the entire ballot--including local offices.
If you intend to live outside the United States indefinitely and do not maintain a domicile in your previous state of residence, under federal law you maintain the right to vote for federal offices (President, Vice President, U.S. Senator, and U.S. Representative). That federal law is called The Uniformed and Overseas Citizens Absentee Voting Act (UOCAVA).
For more details check the laws of your state by going to the State Voting Requirements page. Choose your state, and on the following page select "Overseas & Military Voters" and "Eligibility Requirements."