How are these board of officers appointed? Executive Order?
Would the review themselves be a matter of public record or would it be sealed?
The BCNR has statutory authority from Congress and is appointed through the Assistant Secretary of the Navy (Manpower and Reserve Affairs), with the approval of the SECNAV. It's supposed to be non-partisan and is composed of a panel of active duty, retired and civil service personnel. Staff Judge Advocates General (JAG officers...lawyers) make up the bulk of the members. All are very senior (at least 0-6, but usually Flag or General rank). Usually, the Board consists of not less than three sitting members to hear each individual case.
It's almost like an appellate hearing in the civilian courts, where you appeal your conviction to a higher court (say, for example, to the 5th Circuit Court of Appeals, although this is not a perfect parallel).
You can also appeal an unfavorable outcome at court martial, eventually, to the Court of Military Appeals, which is comprised entirely of very senior Staff JAG officers (kind of like the military version of the Supreme Court). If Kerry had his discharge adjudicated at this level, he did something really serious, which I doubt.
More than likely, he sought to have an upgrade of a General discharge to Honorable. Since he's now staked out this persona as a Captain America war hero-type, he can hardly afford to come off as anything less than perfect to the electorate.