I do not understand... (spoilers)

I don't think Hutton killed Moore. But it was a bit ambiguous when I did see the first film. Assuming Moore killed himself, it must've been because of the inevitability that his career was over. He knew that he may have been a scapegoat for Captain Campbell's murder.

Col. Moore (James Woods) did not kill himself but was murdered by Kent (Timothy Hutton). Presumably, his release from the prison made it easier to kill him. In the film, Kent was at the prison when Brenner went to look for Moore and was told that he was released. By then, Moore had already been killed by Kent. Since Moore at that time was considered the main suspect, Kent had him murdered and made it appear to be a suicide. The general and his sidekick Col. Fowler (the black guy) - who had their own things to hide - were naturally more than happy with that conclusion and so told Brenner that the case was solved.