Which English translation of the Bible does the Orthodox church use? By Orthodox I intend both the Greek Orthodox and the Russian Orthodox, and if there are any differences.
As you probably know, the Greek Orthodox Church and the Russian Orthodox Church are particular jurisdictions of the Eastern Orthodox Church.  What follows I think fairly characterizes all jurisdictions in the United States.

There is no official English translation of the Bible recommended for Orthodox in the United States, but as far as I know the guidance is fairly consistent across jurisdictions:

Old Testament

English translations of the Septuagint are preferred over translations of the Masorah.  While there are a few different English translations of the Septuagint available, generally the translation that is in the Orthodox Study Bible is recommended.  

New Testament

Despite the fact that the New King James Version is included in the aforementioned Orthodox Study Bible, clergy seem to almost always recommend first the King James Version, then the Revised Standard Version.  As far as I have seen, the New Revised Standard Version is universally condemned.

I should note that for Liturgical use there are special translations in the service books.  The New Testament lectionary comprises a Gospel book that is kept at the altar and an Apostol, which contains the lectionary readings from Acts and the Epistles.  I think, but am not sure, that almost all Gospel and Apostol books used in services are based closely on the King James Version.  There are also Old Testament readings - especially Psalms - and these readings are in the service books themselves (i.e. there is no lectionary version of the Old Testament similar to the Gospel book and the Apostol).

All this is not to say that there are other English translations that some Orthodox Christians use.  There is, for example, the 2-volume Orthodox New Testament -based closely on the 1904 Constantinopolitan Patriarchal Text (very similar to the Majority Text) - which includes extensive Patristic commentaries.  There is also a more modern English translation of the Patriarchal Text that has been compiled by Fr. Laurent Cleenewerk.  Fr. Lawrence Farley prefers to use the New American Standard Bible in his "Coffee Cup Commentaries".  I am sure others can enumerate many other versions that are occasionally used by Orthodox Christians.