Let's start with their scripture.
D&C 132: 20 Then shall they be gods, because they have no end; therefore shall they be from everlasting to everlasting, because they continue; then shall they be above all, because all things are subject unto them. Then shall they be gods, because they have all power, and the angels are subject unto them.(It is interesting to note that the footnote for the word "gods" includes the Topical Guide reference "Man, Potential to Become Like Heavenly Father.")
37 Abraham received concubines, and they bore him children; and it was accounted unto him for righteousness, because they were given unto him, and he abode in my law; as Isaac also and Jacob did none other things than that which they were commanded; and because they did none other things than that which they were commanded, they have entered into their exaltation, according to the promises, and sit upon thrones, and are not angels but are gods.
Notice how closely the language parallels language that is ascribed only to God and Jesus in the Bible...yet they are referring to mere mortal men here.
D&C 76: 58 Wherefore, as it is written, they are gods, even the sons of God— (speaking of the resurrection of the faithful)
Abraham 4 is particularly disturbing to me. I would encourage reading the whole chapter, but the first verse sets the tone well enough:
1 And then the Lord said: Let us go down. And they went down at the beginning, and they, that is the Gods, organized and formed the heavens and the earth.
At the least this is referring to the polytheistic Godhead, which is contrary to the Bible's very clear monotheistic teachings which are proven by sources like the Dead Sea Scrolls to predate Jesus. It's more likely that the reference to Gods was a council of Gods that Joseph Smith refers to elsewhere.
Here is what I think is one of the most damning evidences of the doctrine of men becoming like God being taught by Joseph Smith. This is the King Follet Sermon.
"God himself was once as we are now, and is an exalted man, and sits enthroned in yonder heavens! That is the great secret. If the veil were rent today, and the great God who holds this world in its orbit, and who upholds all worlds and all things by His power, was to make himself visible,—I say, if you were to see him today, you would see him like a man in form like yourselves in all the person, image, and very form as a man; for Adam was created in the very fashion, image and likeness of God, and received instruction from, and walked, talked and conversed with Him, as one man talks and communes with another.
Here, then, is eternal life—to know the only wise and true God; and you have got to learn how to be gods yourselves, and to be kings and priests to God, the same as all gods have done before you, namely, by going from one small degree to another, and from a small capacity to a great one; from grace to grace, from exaltation to exaltation, until you attain to the resurrection of the dead, and are able to dwell in everlasting burnings, and to sit in glory, as do those who sit enthroned in everlasting power. (My note: in the Bible, only God is enthroned in everlasting power.)
These are incomprehensible ideas to some, but they are simple. It is the first principle of the gospel to know for a certainty the character of God, and to know that we may converse with Him as one man converses with another, and that He was once a man like us; yea, that God himself, the Father of us all, dwelt on an earth, the same as Jesus Christ Himself did; and I will show it from the Bible.
In the beginning, the head of the Gods called a council of the Gods; and they came together and concocted [prepared] a plan to create the world and people it.
The mind or the intelligence which man possesses is co-equal [co-eternal] with God himself.
They are given to me by the revelations of Jesus Christ; and I know that when I tell you these words of eternal life as they are given to me, you taste them, and I know that you believe them. You say honey is sweet, and so do I. I can also taste the spirit of eternal life. I know that it is good; and when I tell you of these things which were given me by inspiration of the Holy Spirit, you are bound to receive them as sweet, and rejoice more and more."
(Read the full sermon at http://www.utlm.org/onlineresources/sermons_talks_interviews/kingfolletsermon.htm)
These are not isolated teachings. Many are familiar with the couplet that says that, "God once was as man is now, and man may become as God now is."
Here are some further quotes by Bruce R. McConkie, a previous General Authority:
"The Father is a glorified, perfected, resurrected, exalted man who worked out his salvation by obedience to the same laws he has given to us so that we may do the same," (Bruce R. McConkie, A New Witness for the Articles of Faith, Salt Lake City: Deseret Book Company, 1985, p. 64).
"Man and God are of the same race, and it is within the power of righteous man to become like his Father, that is to become a holy Man, a Man of Holiness," (Mormon Doctrine, p. 465-466).
"This Holy Man, the Father of us all, who reigns supreme and is a saved being , ordained and established a plan of salvation so that his Firstborn and all his spirit children might advance and progress, become like him, have all power, know all things, live in the family unit, having eternal increase of their own or in other words, that they might gain for themselves immortality and eternal life," (A New Witness, p. 704).
The Gospel Principles manual, used to teach on Sundays, says this in Chapter 2:
"Our Heavenly Father called a Grand Council to present His plan for our progression (see Teachings of Presidents of the Church: Joseph Smith, 209, 511). We learned that if we followed His plan, we would become like Him. We would be resurrected; we would have all power in heaven and on earth; we would become heavenly parents and have spirit children just as He does."
Chapter 47 says that,
"When we lived with our Heavenly Father, He explained a plan for our progression. We could become like Him, an exalted being."
This has been reflected more recently, such as in D. Todd Christopherson's April 2011 General Conference talk titled "As Many As I Love, I Rebuke and Chasten" where he says that God's purpose is for us "to be with Him eternally, and to become even as He is."
These things have been taught from the pulpit by prophets and published in works written and endorsed by General Authorities. These things were taught by their very first prophet as revelation. The Topical Guide still hints at it. It is still being taught.
In fact, it is a logical outreach of the doctrines the church espouses. If we are the literal children of heavenly parents, well, how did they become heavenly parents? And if children grow up to become like their earthly parents, as far as being adults and hopefully successful and productive citizens and parents, will we not "grow up" to become like our heavenly parents as far as having the opportunity to reach the same or similar status and have our own spiritual families as well? Moreover, the very fact that Jesus, who Mormons teach is our literal spirit elder brother, was able to become a God would argue that we could, too, and that that is the whole point of God's plan.
This is what Mormon authorities have taught explicitly in the past, and even if they're far more shy about teaching it so openly now--and even if they are sometimes outright duplicitous in denying that the doctrine is taught, as Hinckley did to the public even though he himself taught it within the church--it doesn't not change its existence or that it is the logical outgrowth of the false doctrines about God and how we came to be that the Mormon church teaches.