How does the LDS view of achieving godhood compare with the early church view of Theosis?

They are separate doctrines with a few similarities. The similarities should not be used to validate or invalidate the distinct doctrines.

Both doctrines teach that a believer can take on certain characteristics of divinity. Neither would have the ascendant believer supplant God the Father as supreme. This is, largely, the end of the similarities.

The LDS doctrine, called "Exaltation" by Joseph Smith, states that a person can become like God in essence or nature. It takes advantage of texts which imply we can become like God or transformed into something like Him, but also passages which call us sons or imply divine begetting of us.

The LDS doctrine also includes the idea that God the Father went through this process Himself at some point and is, therefore, no more (or less) necessary and eternal than are we mortals.

The Eastern Orthodox idea of Theosis is generally understood to mean that we take on certain characteristics of divinity at some point. These characteristics may include aspects such as immortality, incorruptibility and perfect communion with God. There is some variation, as far as I can see, in the Eastern Orthodox position with regard to whether this means we become truly, independently divine. For example, Leonid Ouspensky writes that "By these energies [characteristic of the common nature of the three divine persons], man surpasses the limits of the creature and becomes a ‘partaker of the divine nature’." Others make a strong point to affirm that God is unique and transcendent.

In any event, Eastern Orthodoxy would in no way affirm the possibility that God the Father underwent some change from mortal to immortal, or that He is not divinely necessary or self-sufficient from eternity-past.

Here is a sampling of key Patristic quotes on Theosis:

Irenaeus of Lyons: (2nd century)

our Lord Jesus Christ, the Word of God, of His boundless love, became what we are that He might make us what He Himself is (Against Heresies Book 5, Preface)

We believe the Savior can make us as He is (e.g. Moroni 7:48)

Clement of Alexandria: (2nd century)

the Word of God became a man so that you might learn from a man how to become a god (Exhortation to the Greeks, 1)

We believe that we not mere creations, but veritable children of God, who can become as our Father is.

Origen of Alexandria: (3rd century)

It was by the offices of the first-born that they became gods, for He drew from God in generous measure that they should be made gods, and He communicated it to them according to His own bounty. The true God, then, is "The God," and those who are formed after Him are gods, images, as it were, of Him the prototype. (Commentary on John Book II, chapter 2)

We believe we are created in the image and likeness of God with the capacity to become like Him.

Athanasius of Alexandria: (4th century)

For the Son of God became man so that we might become God (On the Incarnation of the Word 54.3)

Through His atoning sacrifice Jesus Christ does not merely provide a way to get us back to how things were before the Fall, but to take us far beyond that point, to make us joint-heirs with Him (see further discussion of the relevant theology on my channel here).

Augustine of Hippo: (5th century)

But he himself that justifies also deifies, for by justifying he makes sons of God. ‘For he has given them power to become the sons of God’. If then we have been made sons of god, we have also been made gods.(On the Psalms 50.2)

This concept of deification is consistent with the teachings in Doctrine & Covenants 132:20.

(many other relevant Biblical & Patristic citations are available here)

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All of the above are consistent with what Latter-day Saints believe. Of course, Latter-day Saints also believe more about Theosis than is preserved by the Patristic writers (our belief in Theosis is based on modern revelation, not solely deduction from ancient writings)

A similar point could be made with respect to modern declarations of belief in many faiths: they tend to be much lengthier and go into far greater detail than the ancient texts that introduce the matter (e.g. virtually all commentaries on the Gospel of Matthew are longer than the Gospel of Matthew).

As a Latter-day Saint, I do not feel a need to explain away the straightforward meaning of these passages. I believe what these Patristic citations say about Theosis; I also believe a lot more on the topic. I believe God knows more on the matter than is found in the passages preserved by Irenaeus, Clement, and others, and that God has spoken through modern prophets to reveal a portion of that knowledge.

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Conclusion

The teachings of the Church of Jesus Christ of Latter-day Saints on Theosis are based upon modern revelation, not deduced from the Patristic teachings. However, there is nothing in the Patristic citations above that is not in harmony with what Latter-day Saints believe today.

^{Disclaimer: these thoughts are the product of my own study and do not constitute official statements by the Church of Jesus Christ of Latter-day Saints}