The custom is to wash one’s hands three times, alternating between hands, after attending a funeral.(Mishnah Berurah 4:43)
The Sefer Ishei Yisreol (page 28) writes that according to the Pri Megadim and Chochmas Adam one is not required to wash hands after a funeral unless he was within four Amos of the corpse. However, the custom of the Chazon Ish was to wash his hands upon attending a funeral even if one was not within four Amos of the corpse.
One does not take the vessel from another person’s hand and does not hand the vessel to another person. Rather one places it down and the next person takes it. (See Gesher Hachaim ibid.) The custom is to place the vessel upside down. (See Netai Gavriel Aveilus chapter 70:2)
The poskim write that the custom in Yerushalayim is that one does not dry his hands with a towel and rather lets them dry on its own. (See Gesher Hachaim page 93) The Kaf Hachaim (4:78) cites those who are lenient to use a towel during the winter when it is cold. Harav Shlomo Zalman Auerbach zt”l (cited in Ishei Yisroel) explains that one only refrains from drying his hands when leaving a funeral. However, when washing one’s hands after leaving a cemetery one may dry his hands with a towel.
One is not to enter into a house until after the washing (Rama cited by Mishnah Berurah ibid.). (Some allow entering into a Beis Midrash. [Nitei Gavriel 70/5 and so is the Chabad custom])