How does Dietrich Bonhoeffer's theology relate to salvation by faith alone?

My answer is a bit long, but I hope it helps.

There are a number of tensions in Lutheranism (Lutherans tend to like theological tensions), some of which I think are pertinent here. Mainly: Law/Gospel and Justification/Sanctification.

The Law is God's Word which demands; summarized by Christ as Love the Lord your God with all your heart, soul, and mind, and your neighbor as yourself. The Gospel is God's Word which forgives on account of Christ; i.e. God's forgiveness for our failure to fulfill His Law.

The Law also has three functions: (1) restrain outward sin, (2) drive us to seek God's forgiveness through the Gospel, and (3) show us what is pleasing to God.

Related to this is the concept of Justification and Sanctification. God justifies us freely in His sight through Christ; we receive his righteousness as a gift through faith. Then, once justified, God works through us to sanctify us to conform us more and more to Christ's image. Ephesians 2:8-10 provides a nice summary; verses 8 and 9 basically cover Justification, while 10 covers Sanctification.

The point, then, is that if we are made Christ's people (i.e. we receive Christ's righteousness through faith), then we will want to do the things which please God (i.e. the Law). We do them not to "earn" our salvation/justification (because salvation is a gift), but rather because that's just "what we do" as Christ's people (i.e. we're being Sanctified).

Jesus' image of himself as the vine and his disciples as the branches is relevant here (John 15): we produce good fruit as Jesus' disciples, because we are grafted onto the vine through faith.

So, that brings us to Bonhoeffer: "cheap grace is the preaching of forgiveness without requiring repentance, baptism without church discipline, Communion without confession, absolution without personal confession. Cheap grace is grace without discipleship, grace without the cross, grace without Jesus Christ, living and incarnate."

To receive the Gospel as Good News, I must first realize that I am a sinner. That is, the Law must be preached to me in order to call me to repentance. Otherwise, the Gospel is not "Good News," but simply reaffirms my sinful behavior; i.e. it's "cheap grace." That is, if I believe that I am not a sinner, then being told that I am forgiven means nothing to me; I need the Law in order to appreciate the Gospel. I need to know that my forgiveness actually cost something; it cost Christ's life. God has already forgiven me through Christ, but in order to know and really understand what this means, I need to know that I actually need this forgiveness.

To be Baptized means to be re-born into the image of Christ and to begin the process of sanctification. Luther called this a daily dying and rising, though; that is, I will not be perfect in this life, so my entire life is one of repentance. At times I need someone to point out my sin and rebuke me (e.g. church discipline) so that I do not fall away from the faith.

To receive Communion (i.e. to commune with Christ and his people at the altar), I need to recognize that I go to the altar not because I am worthy to be there, but because Christ has brought me there and reconciled me to him and to the rest of the Church. I confess my sins and my unworthiness (i.e. I feel the effect of the Law) and I receive forgiveness and reconciliation (i.e. the Gospel).

To receive absolution as the forgiveness that it is, I need to confess that I am a sinner (similar to what I said above).

Finally, if I truly have faith, then I will want to do things which are pleasing to my Heavenly Father and be with Christ and his people.

I've read Bonhoeffer's works and his emphasis tends to be on getting the Church to actually "be the Church." That is, if we truly believe, then we ought to live as Christ has called us to live.