The Halachot of how to put on a Tallit are outlined in Shulchan Aruch, Siman 8. There, Maran, based on his opinion in the Bet Yosef, does not require wrapping the Tallit in a manner known as "Atifat Yishmaelim," the way Arabs wrap their headdress. The Geonim disagreed and required "Atifat Yishmaelim," which entails first putting the Tallit over the head and then gathering the ends and throwing them over the shoulders. The accepted custom is to take into account the opinion of the Geonim and do "Atifat Yishmaelim."
However, there is a common misconception as to how exactly to wrap like "Atifat Yishmaelim." Many people mistakenly think that the Tallit should be worn over the head, covering the entire face. This is clearly not the way Arabs wear their Kufiyahs. They wouldn’t be able to see anything. Rather, "Atifat Yishamelim" requires placing the Tallit over the head and wrapping it around the bottom of the face at the level of the chin, leaving the face uncovered. The Poskim say that one should leave the Tallit covering his head in such a fashion for at least the time it takes to walk four Amot, which is equivalent to three seconds.
The question arises whether one may answer Kaddish while in the middle of wrapping the Tallit. This touches on a general principle, by which if someone has already recited the Beracha on a Misva, but not yet begun to perform the act of the Misva, answering Kaddish would be a "fatal interruption." However, if one already began the act of the Misva and is now involved in completing the Misva, he may answer the Kaddish. Therefore, if he has not yet even put the Tallit on his head, he may not answer, but if he already has placed it on his head and is now involved in wrapping "Atifat Yishmaelim," he may answer.
There is a major disagreement as to how exactly to wrap the Tallit. The Shalmeh Sibbur )R.Yisrael Yaakov Algazi, 1680-1757, Turkey-Jerusalem) holds that first, a person wraps his head together with his entire body in the Tallit, and then he gathers it and wraps it around his head and shoulders. The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) held that from the beginning, the Tallit is gathered together around the head and neck, and only then is it released to cover the entire body. While both opinions are considered legitimate, Hacham Ovadia would put on his Tallit in accordance with the Ben Ish Hai.
Hacham Ovadia points out that according to the practice of the Ben Ish Hai, those who have the custom to recite the Pesukim of "Ma Yakar…Yirvayun B’Deshen… etc." should not do so until after they release the Tallit to cover the body. Until that point, one is still performing the initial act of the Misva and may not interrupt with these "extra" Pesukim, whereas answering Kaddish is permitted, as it is a critical Tefila. Hacham Ovadia himself would not verbalize these Pesukim at all. Rather, he would think them while wrapping the Tallit around his head, and naturally this would not be an interruption of the Misva. However, according to the method prescribed by the Shalmeh Sibbur, where the tallit is first wrapped around the body, reciting the Pesukim would not be an interruption during the wrapping of the head, because the prime act of the Misva has already been performed.
The Tallit should be put on like "Atifat Yishmaelim," which means to cover the head and wrap the Tallit up to the chin, for at least three seconds. One may answer Kaddish while wrapping the Tallit, as long as he already put it around his head.