Biblical theology (a slightly unfortunate expression, because all theology should be biblical!) means ‘looking for, and following, the connecting themes that run through the Scriptures’ (Millar and Campbell).  In other words, it is discerning the Bible’s overall story; its big picture.

How can biblical theology be used to preach Christian truth from the Old Testament?

Gary Millar offers the following options:

1. Follow the plan.  Sometimes, a forwards trajectory is present in the OT passage itself.  Ruth 4 (which traces Ruth’s descendants down to King David) and 2 Kings 11 (where that same royal line is in jeopardy) are cases in point.  The preacher can simply follow where such passage are pointing to.
2. Move to the fulfilment.  The OT contains many promises of God’s ultimate Rescuer, and the preacher may, instead of following the unfolding plan, jump straight to the fulfilment.  Such passages in the ‘king’ laws of Deuteronomy 17, the ‘suffering servant’ teaching of Isaiah 53, and the vision of the son of man in Daniel 7.
3. Expose the problem.  Often, OT passages do not highlight God’s rescue plan, but rather expose human need and sinfulness.   We learn of ‘the inadequacy of the people God chooses to use (whether the proud and stubborn Moses in Exodus 6, or the licentious Samson in Judges 13-16), or the fickleness of God’s people (in, say, Exodus 32), or their downright wickedness (take Judges 21, for example).’  These illustrate fundamental problems of the human situation, and the preacher can then show how God has dealt with then through Jesus Christ.
4. Highlight the divine attribute.  Instead of (or as well as) focussing on human failure, the preacher may put the spotlight on how God responds.  Both can be done from Exodus 32-34, which tells both of how God’s people have messed things up and also of his gracious character (Ex 34:6).  God’s grace is equally prominent in Judges 13-16.  From Ex 6 a contrast can be drawn between the character of Moses and that of God.
5. Focus on the action.  Here, the preacher focusses not so much on who God is as what he does.  From Exodus 7-11 the preacher can not simply state that God is all-powerful, but state how he defeats the powers of evil.  Similarly, in the David and Goliath story in 1 Samuel 17, the preacher can emphasise the way in which God takes on his enemies (cf. Col 2:15).
6. Explain the category.  Various OT passage introduce or illustrate key biblical themes – substitution in Gen 22, divine justice in Deut 20, divine faithfulness in How 1-2.  The preacher can demonstrate and develop such themes.
7. Point out the consequences.  Time and again, OT narrative demonstrate the consequences of living, or not living, according to God’s will.  This can be taught from Josh 7, or the Bathsheba incident in 2 Sam 12.  The wisdom literature can also be handled in this way.
8. Describe the ideal human character.  Passage such a Psa 1, Prov 3 and Mic 6:8 celebrate the character of a godly person.  Rather than simply moralising, by saying, ‘We ought to be like that’, the preacher can point forward to the character of Jesus Christ, and to what God’s people will be like in the new creation.  (Such an approach can also be taken when preaching from the Sermon on the Mount and other NT passages).
9. Satisfy the longing.  Some OT passage express deep pain and longing.  Examples include Neh 9 and 13, Dan 9 and many of the Psalms (including the troublesome Psa 137).  In such parts of the OT, old covenant believers are longing for the new covenant.  The preacher can move from disappointment to resolution by pointing to Jesus.

in Millar and Campbell, Saving Eutychus, ch. 5.  Also online here.