Who were the men 'of reputation' Paul met with in Jerusalem?

Paul met privately with the recognized leadership of the Jerusalem church. That group included Peter and the eleven, and possibly a few other Jewish elders. They were the men 'of reputation'—the church's leadership—who heard Paul explain the gospel he preached to the Gentiles.

Les explains that the men 'of reputation' were the leadership of the Jerusalem church. He identifies them as Peter and the eleven, and possibly a few other Jewish elders. Paul met with them privately in a small ante room, along with Barnabas. Paul communicated the gospel he preached among the Gentiles—a message that left out circumcision and law keeping. These Jewish leaders could not comprehend that. The context then notes Titus, a Greek convert with Paul, was not compelled to be circumcised, showing they held them at bay at that council meeting.

Galatians 2:2

And I went up by revelation, and communicated unto them that gospel which I preach among the Gentiles, but privately to them which were of reputation, lest by any means I should run, or had run, in vain.

What did Paul and Barnabas try to prove to the leadership in Jerusalem?

Paul and Barnabas sought to demonstrate to the Jerusalem leadership that nothing may be added to the relationship between Christ and the believer except faith, and faith alone in Paul's Gospel. They came from Antioch to establish this point, showing that salvation rests entirely on faith rather than additional requirements.