Hallel is recited on all eight days of Hanukah, with a Beracha, even in a house of mourning, Heaven forbid. Even the mourner himself recites Hallel, with a Beracha, on each day of Hanukah.
It is customary to recite the 30th chapter of Tehillim – "Mizmor Shir Hanukat Habayit Le’David" – toward the end of the prayer service on each day of Hanukah, as this was the Psalm sung by the Leviyim each day during the eight-day rededication of the Bet Ha’mikdash at the time of the Hanukah miracle. Some communities have the custom to recite this chapter immediately after the Shir Shel Yom; after reciting "Bet Yaakob," they recite the daily Shir Shel Yom and then "Mizmor Shir Hanukat Ha’bayit." Others, however – and this was the practice in Halab (Aleppo, Syria) – recite the Psalm at the time the Sefer Torah is returned to the Hechal, immediately after reciting the verse of "Yehalelu."
During the eight days of the Temple’s rededication, the Leviyim sang only the Psalm of "Mizmor Shir Hanukat Ha’bayit," and not the standard Shir Shel Yom. Therefore, when we introduce the Shir Shel Yom on Hanukah, we omit the usual text of "She’ha’Leviyim Hayu Omerim Al Ha’duchan." We recite the first section of the introduction – "Hayom Yom…Be’Shabbat," in order to fulfill the Misva of remembering Shabbat each day (an obligation indicated by the verse "Zachor Et Yom Ha’Shabbat Le’kadesho"). But we omit mention of the Leviyim, since during Hanukah the Leviyim sang only the special Psalm for the Temple’s rededication, and not the standard daily Psalm. This is the ruling of the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), as well as Hacham Ovadia Yosef, as mentioned in Yalkut Yosef (p. 200).
Some Siddurim note that when one recites "Aromimcha Hashem Ki Dilitani" before "Hashem Melech" in the morning (as is the custom of the Sepharadim), he should add during the Hanukah the first verse of that Psalm – "Mizmor Shir Hanukat Ha’bayit Le’David." In truth, however, this is incorrect. There is no need to add this verse at that this point in the prayer service, since we in any event recite the entire chapter later, as mentioned. Therefore, even during Hanukah, we recite "Aromimcha Hashem Ki Dilitani" before "Hashem Melech" without the first verse of "Mizmor Shir Hanukat Ha’bayit."
Summary: Hallel is recited with a Beracha each of the eight days of Hanukah, even in a house of mourning, and even by a mourner himself, Heaven forbid. Each day of Hanukah we add the Psalm of "Mizmor Shir Hanukat Ha’bayit"; some recite it after the Shir Shel Yom, while others recite it as the Torah is returned to the Hechal. We do not add the first verse of this Psalm before reciting "Aromimcha Hashem Ki Dilitani" before "Hashem Melech." When reciting Shir Shel Yom during Hanukah, we omit the phrase "She’ha’Leviyim Hayu Omerim Al Ha’duchan."
