It's almost a shame that Spock's death occurred what would turn out be so early in the history of the fanchise, as, for this fan's money, it set an incredibly high bar for any deaths to follow.
While there are differing views regarding whether or not it constitutes a character, I feel the death of the Enterprise in TSFS merits inclusion on this list.
Kirk: My god Bones, what have I done?
Bones: What you had to do. What you always do. Turn death into a fighting chance to live.
I'll add the death of David Marcus in the same film, not so much for the death itself, which is tragic but for the fact that it involves someone we don't know very well (though he died trying to save Saavik's life), but for Kirk's reaction. I honestly don't know whether his failure to fall into his chair was intentional or not, but I'll use that scene to combat any criticism of Shatner's acting choices.