Round 1: 1 vs 6, 2 vs 5, 3 vs 4
Round 2: 1 vs 2, 5 vs 3, 6 vs 4
Round 3: 3 vs 1, 2 vs 6, 4 vs 5
Round 4: 1 vs 4, 2 vs 3, 6 vs 5
Round 5: 5 vs 1, 4 vs 2, 3 vs 6
In case of a tie in points, the player with the higher Berger coefficient will be ranked higher (a standard coefficient for round-robin systems; if anyone is interested and cannot find an answer to what it is, I can explain it in more detail). Good luck to everyone!
P.S.: Excel shows how the coefficient is calculated.