Records of General Courts Martial and Courts of Inquiry of the Navy Department are at the National Archives, on microfilm publication M273. Cases for the year 1865 are on rolls 142 to 161.
If you contact the National Archives directly they will advise you which particular roll contains the transcripts for lieutenant King's case, and they will give you a quote for copies.
Callahan shows that acting volunteer lieutenant Edward M. King was honorably discharged in 1867, so the case obviously had no bearing on his career.