In verse 24-25b, Paul proclaims the Gospel in concentrated form.  God set forth Christ as the ἱλαστήριον or "means of atonement" through His blood.  Christ's penal substitutionary death accomplishes redemption through which we are counted righteous before God.  Paul points out the overflowing mercy of God in this by saying it was a "gift by His grace."  This stands in stark contrast with the preceding context which shows that all have sinned and fallen short of the glory of God.  The blood of Christ is the means of our redemption and therefore our justification through faith in Christ.

Paul continues his string of prepositions and declares what this accomplished. I believe there is a progression in these purposes which lead to the final and climactic clause.  All three show that it was to prove God's righteousness.  The first clause shows it needed to be proven because God had formerly passed over sins.  The second clause shows that in the present time the righteousness of God has been manifested because of Christ's work on the cross.

 Finally, the climactic and summary purpose clause (εἰς τὸ εἶναι).  This shows what it means that the righteousness of God has been manifested through the blood of Christ Jesus:  God is declared to be just (δίκαιον) and the One who justifies (δικαιοῦντα).  The fullness of the righteousness of God in both its mercy to those who have faith and its punitive justice against all who have sinned is accounted for through the eschatological manifestation of Christ and His atoning work on the cross.  Therefore, the Old Testament riddle of the merciful God and the God who does not acquit the guilty is answered in the person and work of Jesus Christ.