For my write-up today on Justification and Variegated Nomism, Volume 2: The Paradoxes of Paul, I'll quote what Peter O'Brien says on pages 372-373:
"...it is mistaken to maintain that Paul was boasting simply in his Jewish privileges and status...If Paul was boasting simply in his national privileges, why does he not describe his earlier righteousness as 'our righteousness' (thus identifying with his compatriots) and not 'my own righteousness'...? The context makes it plain that Paul speaks of his own 'righteousness'...because at this point he is drawing a contrast...with his opponents, that is, fellow Jews who shared his 'national privileges,' and showing why he has greater grounds than they for boasting in the flesh."
There are some New Perspectivist authors whom I have not read (i.e., James Dunn), and some whom I have read. A view among many New Perspectivists is that Paul did not believe that Judaism was a religion that promoted people boasting in their own righteous deeds and attempting to earn their own salvation, for Judaism had a notion of grace. When Paul criticizes the Israelites for trying to establish their own righteousness and not submitting to the righteousness of God (Romans 10), or says that he could boast in the flesh (Philippians 3), many New Perspectivists claim that Paul is not presenting Judaism as a religion of self-righteousness in terms of deeds. Rather, for many New Perspectivists, the key issue for Paul was the inclusion of Gentiles into God's people, and so Paul in those passages is criticizing Judaism as nationalistic and exclusive----the Jews are holding on to their national privileges of righteousness within the context of God's covenant with Israel, from which Gentiles are excluded.
But O'Brien argues quite well that Paul is not only talking about national privileges in Philippians 3. Granted, Paul is saying that he could boast in features of God's covenant with Israel----being a Hebrew, being circumcised, observing the Torah, etc. But Paul also appears to be saying that he personally excelled in his Torah observance, which is boasting in one's good works. Does that mean that Judaism (a complex phenomenon in itself) was a religion of self-righteousness? Not necessarily, for it had grace. But some people could have taken elements of Judaism in a self-righteous direction, the way that many Christians have done with Christianity.