The Greek word translated "firstborn" here is πρωτοτοκων (prototokon). This is the plural form of πρωτοτοκως. The word is a compound word from πρωτος (proto-first as in prototype) and τοκως (tokos-born). Literally, then the word means "firstborn [ones]." The word "firstborn" in English could either be singular or plural. In this case, it is meant to be plural. So, what is "The Church of the Firstborn [ones]?" Let's go to D&C 93 for clarification.
"And now, verily I say unto you, I was in the beginning with the Father, and am the Firstborn; And all those who are begotten through me are partakers of the glory of the same, and are the church of the Firstborn." (D&C 93:21-22).
Christ, in the pre-mortal realm, was the Firstborn Son. As such He is heir to all that the Father has. Those who are reborn, "begotten through [Jesus]," become "partakers of the glory of the [Firstborn]" who is Jesus. They, then, become "the church of the Firstborn." The meaning is that those who are begotten through Jesus Christ receive his glory and become heirs with Jesus Christ, they each become "firstborn [ones]" in their inheritance from God.
This doctrine is taught in passages such as Romans 8:14-21, Galatians 4:1-7, 1 John 3:1-3, and D&C 76:24. Through Jesus Christ, the saints become heirs of God as though each of us were the Firstborn Son. We all, through Jesus, can inherit all things (See Matt. 24:47, Romans 8:32, D&C 76:58-59).
"He that overcometh shall inherit all things; and I will be his God, and he shall be my son." (Revelation 21:7).
Unfortunately, this meaning is lost in the English version of Hebrews 12:23. I think most people interpret "firstborn" as being singular and thereby referring only to Jesus.
(It should be clear on careful reading though: "church of the firstborn, which are written").
I believe this number confusion (singular versus plural) is also true of D&C 93:22. Here, I believe, "church of the Firstborn" should not be capitalized since it probably is intended to be understood as plural ("church of the firstborn [ones]") and not as a reference to Christ the Firstborn alone.
I believe this verse would be understood properly thus:
"And now, verily I say unto you, I was in the beginning with the Father, and am the Firstborn; And all those who are begotten through me are partakers of the glory of the [Firstborn], and are [therefore] the church of the firstborn [ones]." (D&C 93:21-22).
Also, this nuance is lost in the foreign language translations of the scriptures. Although in Portuguese and in Spanish translations of the Bible, Hebrews 12:23 reads "Church of the firstborn ones (plural)" in the translations of D&C 93:22 it reads "Church of the Firstborn (singular)." I suspect this problem exists in all of the foreign language translations done by the Church.
Anybody know how to contact the Church translation committee?