Here Paul continues the argument he made in the last section, explaining on theological grounds why Peter's behavior was so problematic. In verses 15 and 16 Paul affirms a basic point of agreement between them that he can use as a starting point for the rest of his argument. Both Peter and he know that it is only by the faithfulness of Christ in his death that one becomes part of God's people. Works of the Jewish Law do not contribute to one's justification. Peter knows that one doesn't need to become a Jew, to live like a Jew including being circumcised and following kosher food laws to be part of God's people. Through the cross God acted to break down the barrier between Jew and Greek showing that Abraham's family was meant to be a worldwide family. All one needs to do is believe to enter that family. At the end of verse 16 Paul alludes to Psalm 143:2 to provide Scriptural proof for his point. No one can keep the law well enough to please God anyways. Christ's work is necessary because of our sin.
15 "We who are Jews by birth and not sinful Gentiles 16 know that a person is not justified by observing the law, but by faith in Jesus Christ.  So we, too, have put our faith in Christ Jesus that we may be justified by faith in Christ  and not by observing the law, because by observing the law no one will be justified.
17 "But if, in seeking to be justified in Christ, we Jews find ourselves also among the sinners, doesn't that mean that Christ promotes sin? Absolutely not! 18 If I rebuild what I destroyed, then I really would be a lawbreaker.19 "For through the law I died to the law so that I might live for God. 20 I have been crucified with Christ and I no longer live, but Christ lives in me. The life I now live in the body, I live by faith in the Son of God , who loved me and gave himself for me. 21 I do not set aside the grace of God, for if righteousness could be gained through the law, Christ died for nothing!" (TNIV)
In the following two verses Paul deals with the charge that would have been leveled against Jewish Christians who ate with Gentiles, namely that eating with sinners means that you condone sin (Jesus was similarly accused when he ate with tax collectors and prostitutes). Paul turns that around and claims the opposite, putting pressure on his opponents. If one of the main purposes of Christ's death was to expand the people of God to a multi-ethnic family, then doing anything to build divisions along ethnic lines, like forcing Gentile converts to follow the law would be an act of sin.
Paul continues on the offensive in the last three verses. His core claim is that requiring law observance invalidates the work of Christ to break down the barriers between Jew and Gentile. All who are in Christ have died to the need to follow the law because they have been co-crucified with Christ. We have been united with Christ in his death and resurrection which frees us from the bondage of sin and thus makes the law no longer the primary means of grace to keep us in good covenantal relations with God. The grace that keeps us in God's family is the grace we receive because of Jesus death on our behalf.
 There is some uncertainty here about the meaning of the phrase rendered by the TNIV as 'faith in Christ.' Many interpreters opt for the 'faithfulness of Christ.' Both are grammatically possible. See Fee (84-88) for a strong defense of the traditional rendering and Longenecker (87) for a concise argument in favor of the latter option. In this particular instance it seems to depend on how repetitive you think Paul is being. Is he expressing the same exact idea in three separate instances in a very short space in 2:16? Fee counters by claiming that it is precisely this repetition that makes the rhetoric effective. What's not commonly looked at is how Paul's allusion to Psalm 143 may help clear up the difficulty. There salvation is clearly based on God's own faithfulness just one verse prior to the one Paul cites. That plus the unlikeliness of the repetition very tentatively pushes me toward the 'faithfulness of Christ' as being the best translation.