The prevalent custom is to recite Selichot during the early morning hours, i.e. at the end of the nighttime hours, before Shacharit prayers. Maran Rabbeinu Ovadia Yosef zt”l writes that the reason for reciting Selichot during the early morning hours is based on the words of the holy Zohar which states that Heavenly kindness is aroused upon the world during the early morning hours, as the verse states, “During the day, Hashem commands His kindness.” However, from the time of Mincha until halachic midnight, the attribute of Heavenly judgment takes control upon the world. From halachic midnight on though, Heavenly kindness and the attribute of mercy are once again awakened and it is for this reason that King David writes, “I shall arise at midnight to thank You.” It is therefore improper to recite Selichot in the beginning of the night which is the time of increased Heavenly judgment. The Mekubalim discuss this matter at length and write that one may not recite Selichot during the first half of the night. It has therefore become the prevalent Jewish custom in all communities to recite Selichot during the early morning hours.
Nevertheless, Maran zt”l writes that those who, for whatever reason, cannot recite Selichot during the early morning hours before Shacharit may indeed recite Selichot before Mincha as well. Although the aforementioned Zohar states that the attribute of harsh Heavenly judgment takes control at the time of Mincha (besides for Shabbat when, on the contrary, the time of Mincha is a time of great Heavenly mercy), it nonetheless seems that Selichot may be recited before Mincha, for the primary hours of the Heavenly judgment’s control are during the actual nighttime hours, i.e. from nightfall until halachic midnight.
It is nevertheless preferable not to recite Selichot during the first half of the night at all, for the great Mekubalim write that one who does so causes great spiritual damage. It is therefore better to abstain from doing so.
Regarding those residing outside of Israel who wish to recite Selichot before halachic midnight in their location but at a time when halachic midnight in Israel has already passed, according to some opinions, they may indeed do so, for this issue is contingent upon the halachic midnight in Israel. Nevertheless, many authorities disagree with this opinion and rule that this matter is indeed contingent on the halachic midnight in one’s present location. This is indeed the final ruling of Maran Rabbeinu Ovadia Yosef zt”l (see Responsa Yechave Da’at, Volume 1, Chapter 46, Chazon Ovadia-Yamim Nora’im, page 4, and Torat Ha’Mo’adim-Yamim Nora’im, page 6). Nonetheless, the great Rishon Le’Zion, Hagaon Harav Yitzchak Yosef Shlit”a writes (in his Yalkut Yosef-Yamim Nora’im, page 38) that there is room for leniency when a Minyan in the United States recites Selichot along with other Jews in Israel via a live hook-up although the time of halachic midnight has not yet arrived in the United States.