Νῦν χαίρω ἐν τοῖς παθήμασιν ὑπὲρ καὶ ἀνταναπληρῶ τὰ ὑστερήματα τῶν θλίψεων τοῦ Χριστοῦ ἐν τῇ σαρκί μου ὑπὲρ τοῦ σώματος αὐτοῦ ὅ ἐστιν ἡ ἐκκλησία
Now I rejoice in [my] sufferings for you, and I fill up in my flesh that which is lacking [as regards] the sufferings of Christ, for [the sake of] His body, which is the church.
What he is doing is clearly a form of penance, i.e. mortifying his body.
The Greek τὰ ὑστερήματα τῶν θλίψεων τοῦ Χριστοῦ literally means "that [which is] lacking in Christ's suffering", however, I think it's very clear from elsewhere in Paul's writings that he doesn't believe Christ's atoning sacrifice, His passion, was incomplete or lacking anything (Romans 3:25; 5:9; 5:18; 2 Corinthians 5:21; cf. 1 John 2:2; 4:10 etc).
He appears to be saying that Christ's sufferings didn't pertain to (didn't 'cover') the kind of sufferings that believers are expected to undergo in general—they will still have to suffer (Christ's suffering doesn't or didn't mean that believers do not have to suffer—Matthew 16:24; Philippians 1:29).
Including various forms of bodily mortification. (Romans 8:13; Colossians 3:5; cf. 1 Corinthians 5:7).
St. Paul writes to the Corinthians:
1 Corinthians 9:27
ἀλλὰ ὑπωπιάζω μου τὸ σῶμα καὶ δουλαγωγῶ μή πως ἄλλοις κηρύξας αὐτὸς ἀδόκιμος γένωμαι
But I punish my body, and bring it into subjection, lest after I have preached to others I myself should be rejected [or, "become reprobate"].
Here he predicates not being himself rejected or reprobate on doing this bodily mortification.
This kind of teaching explains the perennial pratice of mortification of the body among Christians in the form of general penance, fasting and rejection of 'worldly' things.
Some other important and very relevant scriptures are:
Romans 8:16-18 (DRB)
For the Spirit himself giveth testimony to our spirit, that we are the sons of God. 17 And if sons, heirs also; heirs indeed of God, and joint heirs with Christ: yet so, if we suffer with him, that we may be also glorified with him. 18 For I reckon that the sufferings of this time are not worthy to be compared with the glory to come, that shall be revealed in us.
2 Timothy 2:10-13 (DRB)
Therefore I endure all things for the sake of the elect, that they also may obtain the salvation, which is in Christ Jesus, with heavenly glory. 11 A faithful saying: for if we be dead with him, we shall live also with him. 12 If we suffer, we shall also reign with him. If we deny him, he will also deny us. 13 If we believe not, he continueth faithful, he can not deny himself.
2 Corinthians 1:5-6 (DRB)
For as the sufferings of Christ abound in us: so also by Christ doth our comfort abound. Now whether we be in tribulation, it is for your exhortation and salvation: or whether we be comforted, it is for your consolation: or whether we be exhorted, it is for your exhortation and salvation, which worketh the enduring of the same sufferings which we also suffer.
Philippians 3:7-14 (DRB)
But the things that were gain to me, the same I have counted loss for Christ. 8 Furthermore I count all things to be but loss for the excellent knowledge of Jesus Christ my Lord; for whom I have suffered the loss of all things, and count them but as dung, that I may gain Christ: 9 And may be found in him, not having my justice, which is of the law, but that which is of the faith of Christ Jesus, which is of God, justice in faith: 10 That I may know him, and the power of his resurrection, and the fellowship of his sufferings, being made conformable to his death, 11 If by any means I may attain to the resurrection which is from the dead. 12 Not as though I had already attained, or were already perfect; but I follow after, if I may by any means apprehend, wherein I am also apprehended by Christ Jesus. 13 Brethren, I do not count myself to have apprehended. But one thing I do: forgetting the things that are behind, and stretching forth myself to those that are before, 14 I press towards the mark, to the prize of the supernal vocation of God in Christ Jesus.
Far from teaching 'faith alone,' St. Paul teaches a radical, absolutely necessary, partaking in the real "suffering" life of Christ "in order to be glorified with Him." To be 'crucified with Christ' means more than the debt being nailed to the cross, but to be crucified to the flesh, the old man, the world, and the devil. Not in word, but in truth, actually made a saint, "conformed to the image of his Son," "made perfect," and testifies to his personal intention to actively "press toward the mark" to that end—not that somehow God makes you perfect without your co-operation: "Not as though I had already attained, or were already perfect; but I follow after, if I may by any means apprehend, wherein I am also apprehended by Christ Jesus. ... I do not count myself to have apprehended. But one thing I do: forgetting the things that are behind, and stretching forth myself to those that are before, I press towards the mark, to the prize of the supernal vocation of God in Christ Jesus."