Translation:

1 Παῦλος δοῦλος Χριστοῦ Ἰησοῦ, κλητὸς ἀπόστολος ἀφωρισμένος εἰς εὐαγγέλιον θεοῦ,
Paul, a slave of Christ Jesus, called an apostle having been set apart with respect to the gospel of God,

2 ὃ προεπηγγείλατο διὰ τῶν προφητῶν αὐτοῦ ἐν γραφαῖς ἁγίαις
which He promised beforehand through His prophets in the holy Scriptures

3 περὶ τοῦ υἱοῦ αὐτοῦ τοῦ γενομένου ἐκ σπέρματος Δαυὶδ κατὰ σάρκα,
about His son, the one born from a seed of David according to the flesh,

4 τοῦ ὁρισθέντος υἱοῦ θεοῦ ἐν δυνάμει κατὰ πνεῦμα ἁγιωσύνης ἐξ ἀναστάσεως νεκρῶν, Ἰησοῦ Χριστοῦ τοῦ κυρίου ἡμῶν,
the one declared son of God in power with respect to the Spirit of holiness from the resurrection of the dead, Jesus Christ our Lord,

5 διʼ οὗ ἐλάβομεν χάριν καὶ ἀποστολὴν εἰς ὑπακοὴν πίστεως ἐν πᾶσιν τοῖς ἔθνεσιν ὑπὲρ τοῦ ὀνόματος αὐτοῦ,
through whom we received grace and apostleship for the obedience of faith with reference to all the Gentiles for the sake of His name,

6 ἐν οἷς ἐστε καὶ ὑμεῖς κλητοὶ Ἰησοῦ Χριστοῦ,
among whom you also are called of Jesus Christ,

7 πᾶσιν τοῖς οὖσιν ἐν Ῥώμῃ ἀγαπητοῖς θεοῦ, κλητοῖς ἁγίοις, χάρις ὑμῖν καὶ εἰρήνη ἀπὸ θεοῦ πατρὸς ἡμῶν καὶ κυρίου Ἰησοῦ Χριστοῦ.
all the ones residing in Rome beloved of God, called holy ones, grace and peace to you from God our Father and the Lord Jesus Christ.

Commentary:

v. 1: While δοῦλος can be translated “servant” (see ESV, KJV, NASB, NIV, and NRSV) I side here with the NET that chooses to translate it with the stronger “slave”. ἀφωρισμένος is a perfect, passive participle indicating a particular time, in the past, that still effects the present, where Paul was set aside by someone other than himself for the sake of the gospel. This may very well be referring to his conversion on the road to Damascus. The gospel “of God” should be read as a genitive of reference (the gospel “with respect/reference to God” or “about God”).

v. 2: This prepares the reader for Paul’s Christological hermeneutic. He read the Hebrew Scriptures through the eyes of the “Christ-event”. As with the disciples on the road to Emmaus in the Third Gospel, so here it is to be shown that Messiah should be seen throughout the holy Scriptures.

v. 3: The “gospel of God” is “about His Son”. The Son is τοῦ γενομένου, which shows Paul affirms the actual, human origins of the Son of God. The “son” aspect here has less to do with the second person of the Trinity than it does the idea of kingship, or being Messiah. This is made evident by connecting Jesus’ birth with the fact that he came “from a seed of David” (hence, qualifying him to fulfill the promises to David regarding his descendant). κατὰ σάρκα is Paul’s clarification that his birth was in accordance with his human nature. It could be argued that Paul here hints at a high Christology since he has to clarify this point.

v. 4: τοῦ ὁρισθέντος υἱοῦ θεοῦ ἐν δυνάμει allows for us to continue reading this passage as being in reference to Jesus’ royal calling. Christ is given a coronation ceremony as king through the resurrection (for more on this idea read J.R. Daniel Kirk’s Unlocking Romans, pages 40-55). See also this post. πνεῦμα ἁγιωσύνης is a Hebraism for “Holy Spirit”. The Spirit is seen as one who empowers Christians to obtain resurrection life and sonship after the model of Christ.

v. 5: εἰς ὑπακοὴν πίστεως ἐν πᾶσιν τοῖς ἔθνεσιν clarifies Paul’s mission–namely, to bring Gentiles under the rule of Messiah (see Psalm 2). The obedience is that “of faith”, which previews the discussion regarding the faith/works divide.

v. 6: Daniel Wallace has suggested that it is possible to read this verse as saying “called by Christ Jesus if read as a genitive of agency. The vague “of” here is a safer translation in my opinion (Greek Grammer: Beyond the Basics, pg. 127).

v. 7: κλητοῖς ἁγίοις either reads “holy ones” or more traditionally “saints”.