One is required to recite the complete Hallel service on each of the eight days of Hanukah. When reciting the complete Hallel in the synagogue, it is preferable not to recite the introductory Beracha of "Li’gmor Et Ha’hallel," and to instead listen to the Hazzan’s recitation of the Beracha and then answer "Amen." Instinctively, one may have thought that it is preferable to personally recite the Beracha, rather than listen to its recitation by somebody else. In truth, however, the principle of "Be’rob Am Hadrat Melech" establishes that Misvot performed jointly by large numbers of people bring greater honor to the Almighty. It is therefore preferable for the entire congregation to listen to the Hazzan’s recitation of the Beracha, all together, and then recite "Amen."
Nevertheless, a person who did not hear the Hazzan’s recitation of the Beracha may recite the Beracha himself. It occasionally happens, for example, that the Kohanim are returning to their places after Birkat Kohanim as the Hazzan recites the Beracha over Hallel, and they cannot hear every word of the Beracha, or they cannot have the required concentration. In such a case, or in any situation where somebody did not hear the Beracha or could not properly concentration on the Beracha, one may recite the Beracha himself. If an individual recites the Beracha and completes it together with the Hazzan, he should not answer "Amen" to the Hazzan’s Beracha. Although it is permissible to answer "Amen" to a Beracha which one hears during the recitation of Hallel, it is forbidden to make any interruption in between the recitation of the Beracha and the start of Hallel. Therefore, if one hears the Hazzan finish the Beracha after he began Hallel, then he may answer "Amen," but he finishes the Beracha together with the Hazzan, he may not answer "Amen." This is the ruling of Hacham Ovadia Yosef A"H (listen to audio recording for precise citation).
One may recite Hallel at any point during the day, from sunrise to sunset, as alluded to in the verse, "From the rise of the sun until it sets, the Name of Hashem is praised" (Tehillim 113:3). If, however, one recited Hallel before sunrise, he has nevertheless fulfilled the obligation of Hallel, provided that he recited it after Amud Ha’shahar (daybreak, approximately one hour prior to sunrise).
Hacham Ovadia Yosef A"H addressed the question as to whether one who forgot or was unable to recite Hallel the entire day until after sunrise may still fulfill the Misva. The period of Ben Ha’shemashot (twilight), which extends for approximately thirteen minutes after sundown, is treated as a period of "Safek" (doubt), as we are unsure whether to define this period as daytime or nighttime. Hacham Ovadia notes that since the obligation of Hallel was enacted by the Sages, and does not constitute a Biblical obligation, we may assume the lenient possibility – that Ben Ha’shemashot is considered daytime – and allow reciting Hallel during this period. He therefore advises that one who did not recite Hallel before sundown should recite Hallel during the period of Ben Ha’shemashot, but without reciting a Beracha. Thus, for example, if a person was ill and bedridden on one of the days of Hanukah, and was unable to recite Hallel the entire day until after sundown, he should recite Hallel without Beracha during the thirteen-minute period following sundown.
The custom of Halab (Aleppo, Syria) was to recite Psalm 30 – "Mizmor Shir Hanukat Ha’bayit Le’David" – each day of Hanukah as the Torah was returned to the ark. This is thus the proper practice to follow for those who observe the ancient customs of Halab.
Summary: It is preferable to listen to the Hazzan’s recitation of the introductory Beracha to Hallel, rather than recite it personally. Hallel may be recited anytime from sunrise to sunset, but one fulfills the obligation even if he recited it as early as daybreak (approx. one hour before sunrise). One who did not recite Hallel the entire day until after sundown may still recite it during the thirteen minutes after sundown, though without a Beracha. The custom of Halab was to recite the Psalm of "Mizmor Shir Hanukat Ha’bayit" each day of Hanukah as the Torah was returned to the ark.
