Although one should generally refrain from reading Tehillim or other portions of Tanach at nighttime, this is permissible on the night of Yom Kippur, as the night and the day of Yom Kippur have the same status. This is mentioned by the work Hemdat Yamim.
The Hemdat Yamim further notes that there is a custom among especially pious people to remain awake throughout the night of Yom Kippur learning Torah, because one hour of Torah study on the night of Yom Kippur is more valuable than a year’s worth of Torah learning. However, one should not adopt this practice if it will hamper his ability to properly pray the next day. It often happens that people follow stringencies which compromise their ability to observe basic Halacha, and this can happen in regard to Yom Kippur, as well. I know people who decided to stay up all night learning on Yom Kippur, and because of this they were falling asleep during the prayer service and could not pray with any sort of concentration or feeling. If one knows that remaining awake throughout the night will hamper his ability to pray the next day, it is much preferable for him to spend some time learning at night and then go to sleep.
The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) writes that if one generally keeps a glass of water by his bed at night in case he gets thirsty during the night, he should not do so on Yom Kippur night, as he may mistakenly drink during the night, forgetting it is Yom Kippur.
The Gemara mentions a brief text that one should recite before going to the restroom, asking the angels to wait for him while he goes into the restroom. Throughout the year, this text should not be recited, because unlike in the times of the Gemara, we are not on the level where we can assume that angels are accompanying us, and it would thus appear arrogant to recite this text. On Yom Kippur, however, when we resemble the angels – and thus we recite "Baruch Shem Kebod Malchuto Le’olam Va’ed" aloud, like the angels – we may, indeed, assume that we are accompanied by angels. Therefore, one should recite this declaration quietly before going into the restroom on Yom Kippur. This is the ruling of the Ben Ish Hai in his work Od Yosef Hai (Vayeseh, 4). The text is: "Hitkadebu Mechubadim Kedoshim Mesharteh Elyon Shamruni Shamruni Azruni Azruni Hamtinu Li Ad She’ikanes Ve’eseh Sheken Darkam Shel Beneh Adam."
Summary: Torah study is especially valuable on Yom Kippur night, and therefore one should spend time learning Torah on this night. It is permissible to read Tehillim and study other parts of Tanach on Yom Kippur. There are those who stay awake the whole night learning, but one should not follow this practice if this will hamper his ability to pray properly the next day. One should recite the "Hitkabedu Mechubadim" text before using the restroom on Yom Kippur, as opposed to the rest of the year, when this text should not be recited.