There is a dispute among the Halachic authorities concerning the propriety of crying during the Rosh Hashanah prayers. The Vilna Gaon (1720-1797) maintained that one should not attempt to bring himself to tears during the Rosh Hashanah prayer service, whereas the Arizal (Rav Yishak Luria of Safed, 1534-1572) held that to the contrary, it is proper to cause oneself to cry while praying on Rosh Hashanah. The Arizal went so far as to say that if one is not moved to tears during the Rosh Hashanah prayers, this reflects a deficiency in his soul.
The Hid"a (Rav Haim Yosef David Azulai, 1724-1806) suggested a kind of compromise position between these two opinions. He said that the Rosh Hashanah prayers should be chanted in a solemn melody that resembles the sound of weeping and whimpering. The verse says in Tehillim (6:9), "For Hashem has heard the sound of my crying," indicating that prayers should be recited in a "sound" that resembles crying.
It should be noted that even according to the Vilna Gaon’s view, if one is suddenly overcome by emotion during his prayers and begins to cry, he does not violate any Halachic prohibition. The Gaon spoke only of knowingly bringing oneself to tears; one need not be concerned, even according to the Gaon, if he experiences a rush of emotion during the Tefila that causes him to cry.
As for the final Halacha, one may follow the Arizal’s opinion and recite his prayers in a manner that inspires him to cry.
