I know the obvious trope is that the LDS believe in "salvation by works", but I am looking for definitive church teaching on the nature of mans justification. Preferably, I'd like it to include the different levels of heaven in relation to justification.
There maybe some difference in meaning of salvation
Salvation from physical death: all people die. But through the atonement/resurrection of Jesus Christ all people will be resurrected-saved from physical death (in this sense all are saved) (1 Corinthians 15:22)
Salvation from sin: to be cleansed from sin through the atonement, an individual must exercise faith in Jesus Christ, repent, be baptized, and receive the gift of the Holy Ghost (Acts 2:37-38)
Individuals cannot be saved in their sins; they cannot receive unconditional salvation simply by declaring a belief in Christ with the understanding that they will inevitably commit sins throughout the rest of their lives (see Alma 11:36–37). However, through the grace of God, all can be saved from their sins (see 2 Nephi 25:23; Helaman 5:10–11) as they repent and follow Jesus Christ.
Eternal life, or exaltation: In the scriptures, the words saved and salvation often refer to eternal life, or exaltation (see Abraham 2:11). Eternal life is to know Heavenly Father and Jesus Christ and dwell with Them forever—to inherit a place in the highest degree of the celestial kingdom (see John 17:3; Doctrine and Covenants 131:1–4; 132:21–24). This exaltation requires that men receive the Melchizedek Priesthood, and that all Church members make and keep sacred covenants in the temple, including the covenant of eternal marriage. If the word salvation is used in this sense, no one is saved in mortality. That glorious gift comes only after the Final Judgment.
Related, faith in Jesus Christ is much more than passive belief. We express our faith through action—by the way we live. (James 2:18) (this is often the misrepresentation of salvation by works) One cannot earn their way into heaven, but we can do our best to follow God's commandments(requirements). See also Articles of Faith V, 17-18, James E Talmage
This ties directly to justification
A person is justified by the Savior’s grace through faith in Him. This faith is shown by repentance and obedience to the laws and ordinances of the gospel. Jesus Christ’s Atonement enables mankind to repent and be justified or pardoned from punishment they otherwise would receive. (D&C 20:29-34)
The celestial kingdom is the place prepared for those who have “received the testimony of Jesus” and been “made perfect through Jesus the mediator of the new covenant, who wrought out this perfect atonement through the shedding of his own blood” (D&C 76:51, 69). To inherit this gift, we must receive the ordinances of salvation, keep the commandments, and repent of our sins. For a detailed explanation of those who will inherit celestial glory, see D&C 76:50–70; 76:92–96.
“All who have died without a knowledge of this gospel, who would have received it if they had been permitted to tarry, shall be heirs of the celestial kingdom of God;
“Also all that shall die henceforth without a knowledge of it, who would have received it with all their hearts, shall be heirs of that kingdom;
“For I, the Lord, will judge all men according to their works, according to the desire of their hearts” (Doctrine and Covenants 137:7–9).
Commenting on this revelation, the Prophet Joseph said, “I also beheld that all children who die before they arrive at the years of accountability are saved in the celestial kingdom of heaven” (Doctrine and Covenants 137:10). To be exalted in the highest degree and continue eternally in family relationships, we must enter into “the new and everlasting covenant of marriage” and be true to that covenant. In other words, temple marriage is a requirement for obtaining the highest degree of celestial glory. (See Doctrine and Covenants 131:1–4.)
Generally speaking, individuals in the terrestrial kingdom will be honorable people “who were blinded by the craftiness of men” (Doctrine and Covenants 76:75). This group will include members of the Church who were “not valiant in the testimony of Jesus” (Doctrine and Covenants 76:79). To learn more about those who will inherit terrestrial glory, see Doctrine and Covenants 76:71–80, 91, 97.
Telestial glory will be reserved for individuals who “received not the gospel of Christ, neither the testimony of Jesus” (Doctrine and Covenants 76:82). .... A detailed explanation of those who will inherit telestial glory is found in Doctrine and Covenants 76:81–90, 98–106, 109–112
Perdition (not a kingdom of Glory-mentioning as it'll probably come up)
Some people will not be worthy to dwell in any kingdom of glory. They will be called “the sons of perdition” and will have to “abide a kingdom which is not a kingdom of glory” (Doctrine and Covenants 76:32; 88:24). This will be the state of “those who know [God’s] power, and have been made partakers thereof, and suffered themselves through the power of the devil to be overcome, and to deny the truth and defy [God’s] power” (Doctrine and Covenants 76:31; see also Doctrine and Covenants 76:30, 32–49).
I believe depperm has adequately answered the question, but I will add what I think is a helpful summary of the doctrine:
Works cannot save us; only Jesus can do that. But salvation by Jesus is conditional on works like baptism and repentance. We do those works because we have faith in Jesus' power to save us.