The Bet Yosef (work by Maran, author of the Shulhan Aruch) writes that one should not mention the word "Het" ("sin") on the festival of Rosh Hashanah, or confess his sins on this day, as this would be inconsistent with the festive spirit of the holiday. The Zohar (principal text of Kabbalistic thought) likewise discourages making confession on Rosh Hashanah.
However, Rabbi Haim Vital (1543-1620) records that his teacher, the Arizal (Rabbi Yishak Luria of Safed, 1534-1572), encouraged confessing one’s sins and repenting on Rosh Hashanah. The Arizal understood that the Zohar discouraged confessing aloud, in an audible voice, but confessing in a soft, inaudible tone is acceptable and even admirable. Particularly, the Arizal taught that one should confess and repent at the time of the Shofar blowing, specifically during the sounding of the first thirty sounds. During these thirty Shofar blasts, the Satan is confounded and off-guard, and thus unable to prosecute against us before the Heavenly tribunal. This is therefore an especially auspicious occasion for confessing one’s sins and begging God for forgiveness, as the Satan is incapable of prosecuting at those moments. The Kaf Ha’haim (Rabbi Yosef Haim Sofer, Baghdad-Israel, 1870-1939) clarifies that one should confess in between the sets of Shofar sounds. The Ba’al Teki’a (person sounding the Shofar) generally pauses in between the sets of "Tashrat" (Teki’a -Shebarim Teru’a – Teki’a), "Tashat" (Teki’a – Shebarim – Teru’a) and "Tarat" (Teki’a – Teru’a – Teki’a). During those pauses it is proper to confess one’s sins, specifying whichever sins he is aware of, and ask God for forgiveness, in a soft, inaudible tone.
Different views exist as to the propriety of crying during the prayers on Rosh Hashanah. The Vilna Gaon (Rabbi Eliyahu of Vilna, 1720-1797) opposed crying on Rosh Hashanah, noting the verse in the Book of Nehemya (8:10) urging that we observe Rosh Hashanah as a day of joy and festivity, and not with tearful repentance ("Ki Hedvat Hashem Hi Ma’uzchem" – "For the joy of God is your source of strength"). By contrast, the Shela (Rabbi Yeshaya Horowitz, 1565-1630) felt it was praiseworthy to cry from emotion during the Rosh Hashanah prayers. The Arizal went so far as to say that if a person does not shed tears during the High Holidays, this signifies a deficiency in the functioning of his soul.
As for the final Halacha, Hacham Ovadia Yosef ruled that it would be inappropriate to intentionally evoke tears on Rosh Hashanah. However, if a person is overcome by emotion as a natural result of sincere prayer, then he certainly should not restrain his tears, for as the Arizal said, this emotional response testifies to the greatness and purity of one’s soul.
Summary: One should not confess his sins and pray for forgiveness on Rosh Hashanah in an audible voice, but it is commendable to confess and beg for forgiveness in a soft, inaudible voice. This is particularly appropriate during the brief pauses in between the units of Shofar sounds. One should not try to evoke tears during the Rosh Hashanah prayer service, but a person who is overcome with emotion and begins crying should not suppress his crying.