When I wept, and chastened my soul with fasting - The words “and chastened” are not in the original. The literal translation would be, “And I wept (away) my soul with fasting;” that is, I gave myself so much to fasting accompanied with weeping, that my strength was exhausted. This refers to his acts of devotion; to his endeavors to discipline his soul so as to lead a strictly religious life.
That was to my reproach - This may either mean that they accused him of hypocrisy and insincerity; or, that they charged him with folly for being so religious, so strict, so self-sacrificing, so serious - perhaps they would say, so superstitious, so gloomy, so fanatical. The latter best accords with the connection, since it was for his “religion” mainly that they reproached him, Psalms 69:7-9.