What becomes of those that never hear of Christ?

What becomes of those that never hear of Christ? The short answer is that they don’t inherit the kingdom of God. They shall be judged as those "who have sinned without the law" (Rom. 2:12). In the providence of God who is holy, just, righteous, and always without error, those that never hear of Christ don’t go heaven.

What about a longer answer to your question?

While this doesn’t apply to all inquirers, some individuals ask such a question in hopes of locating a personal loophole to escape the eternal torments of Hell. However, it doesn’t work this way! C.S. Lewis in Mere Christianity wrote, "If you are worried about the people outside [of Christianity], the most unreasonable thing you can do is to remain outside yourself." Therefore, the most logical behavior of a person asking such a question is to repent of their own sins and be saved and share the Gospel of Christ in as many ways and with as people as possible.

Why? Because a life without knowing Christ means an eternity in a conscious eternal Hell. 2 Thess. 1:8-9 states that at the last day God “will punish those who do not know God and do not obey the gospel of our Lord Jesus. They will be punished with everlasting destruction and shut out from the presence of the Lord and from the glory of his might.” So, unless one knows Christ - not just knows “of” him, but truly knows him and has a personal minute by minute relationship with him - their last [eternal] days are already written.

So, another question then naturally arises, how can I know Christ? The answer is by being born again (John 3:1-8). Speaking of Jesus Christ, Acts 4:12 states, “And there is salvation in no one else, for there is no other name under heaven given among men by which we must be saved.” In relation to your question this verse states that (1) a person “must be saved” and (2) this is possible only through Christ alone.  

How is this done? When the Holy Spirit convicts someone of their sins then as it states in Romans 10:9-11 those being moved in such a way by the Spirit will:

confess with [their] mouth that Jesus is Lord and believe in [their] heart that God raised him from the dead, [they] will be saved. For with the heart one believes and is justified, and with the mouth one confesses and is saved. For the Scripture says, “Everyone who believes in him will not be put to shame.”

[Jesus] is “the way, the truth, and the life. No one comes to the Father except through [him] (John 14:6). To know Christ in this way means experiencing a supernatural transformation which enables one to be continuously faithful to Christ and his Gospel. This doesn’t mean one will be sinless (cf. 1 John 1:9), but rather in the continuous strength of Christ, his Spirit, his Word, and continuous sanctification they will sin less and less.