Personally, I believe the doctrine of becoming like God is one of the central tenets of the Bible. It is taught in one way or another over and over and over again there - in both the Old and New Testaments. It is my absolute favorite teaching of the Restored Gospel - and, ironically, it is one of the core ties we have to many Eastern religions.
I wrote a fairly long paper at one point for a divinity school class, but here is the bare-bones, stripped down, bullet-point version - to the best of my memory, and not taking the time to quote actual scriptural verses:
1) It is clear that the Bible teaches we are created in the image of God. That is so widely accepted to be a given, although many have interpreted it in varying ways to make it non-literal.
2) It is clear that the Bible teaches that we are to become like God. In the OT, this generally is phrased as follows: "Be __________, because God is __________." The direct line reasoning is that we are to develop a certain characteristic specifically because God has that characteristic. The penultimate statement of this is the Sermon on the Mount, where Jesus laid out traits that make us "blessed" and then says, "Be ye therefore (by the pattern laid out in the previous verses) perfect (complete, whole, fully developed), even as your Father which is in heaven is perfect." In the Book of Mormon version, he makes it even clearer that this is a final state of progression, but he adds himself to the injunction.
3) The NT takes the OT admonitions and actually adds a stated reward. The joint-heir change to the inheritance custom - the "see him as he is, for we shall be like him" - the "one as we are one" - etc. all provide context for the command.
Interestingly, the Book of Mormon says exactly nothing about becoming like God, except in 3 Nephi. I think this is for two primary reasons:
1) The basic teaching is almost omni-present in the Bible, and Mormon and Moroni made it perfectly clear that one of the core purposes of their record was to inspire those who would "believe this" (the Book of Mormon) to "believe that" (the Bible). (Note: It's not to believe IN the Bible, but to actually believe what it says. - hat tip to Robinson) If the concept of becoming like God is central to the Bible, it wouldn't be necessary in the Book of Mormon abridgment to "waste space" detailing it.
2) If it were taught in the Book of Mormon, it would be much easier for non-Mormon Christians to dismiss it as a uniquely Mormon heresy. They still can reject it as such, but the fact that it's not taught in the Book of Mormon means they are rejecting the Bible, not the Book of Mormon, when they reject the concept.
Summary: I see the concept of becoming like God to be a core Christian doctrine - in fact, THE core Christian doctrine of the Bible. I see the rejection of it as THE core abomination of the Great Apostasy.