Halacha requires reciting the full Hallel, with the introductory Beracha, on each of the eight days of Chanukah. Chacham Ovadia Yosef, in his work Chazon Ovadia (Laws of Chanukah, p. 213), rules that women are exempt from the obligation to recite Hallel during Chanukah. A woman who nevertheless wishes to recite Hallel during Chanukah certainly may do so, but she may not recite the Beracha.
The full Hallel is recited with a Beracha during Chanukah even in a house of mourning, Heaven forbid; both the mourner and those praying in his home recite the full Hallel with the Beracha on each of the eight days of Chanukah (Chazon Ovadia, Laws of Chanukah, p. 217).
It is forbidden to speak during the Hallel recitation, with the exception of answering "Amen" upon hearing a Beracha, or responding to Kaddish, Kedusha or Barechu (Chazon Ovadia, p. 221).
The congregation should preferably listen to the Chazan's recitation of the introductory Beracha, answer "Amen," and then begin the Hallel. If a person recited the introductory Beracha himself, he must then begin the Hallel immediately and not make any interruption in between the Beracha and the actual Hallel. If after reciting the Beracha he hears the Chazan or the person standing next to him recite the Beracha, he should not answer "Amen" unless he had begun reciting the Hallel.
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