Reciting Pirkei Avos:
It is accustomed to recite Pirkei Avos on every Shabbos between Pesach and Shavuos. Some are accustomed to continue reciting it after Shavuos, throughout the summer months, until Rosh Hashana. [Practically the Chabad custom is to say Pirkei Avos until Rosh Hashana. On the Shabbosim which are in approximation to Rosh Hashanah, two chapters of Pirkei Avos are read, in order to complete the cycle prior to Rosh Hashanah.]
Reciting “Kol Yisrael…” prior to each chapter and “Rebbe Chanania Ben Akashya…” at the end of each chapter: One is to recite the Mishnaic saying of “Kol Yisrael…” prior to beginning the chapter of Pirkei Avos. At the conclusion of the chapter one is to recite the teaching of “Rebbe Chanania Ben Akashya..”. [On the Shabbosim which are in approximation to Rosh Hashanah, in which two chapters of Pirkei Avos is read, some write one is to read the above opening and closing statements before and after each individual chapter. Others write it is only to be said once; the opening statement at the beginning of the first chapter and the concluding statement after concluding the second chapter. Others write that by the first 4 chapters that are said together [1-2 and 3-4], one reads the opening statement at the beginning of the first chapter and the concluding statement after concluding the second chapter. However by the last two chapters [5-6] one reads the opening statement at the beginning of the first chapter and the concluding statement after concluding the second chapter.]
Learning one Mishna in depth each Shabbos:
It is proper to learn in depth at least one Mishna of the weekly chapter in Pirkei Avos. One is to learn this Mishneh together with its commentaries.
In Eretz Yisrael, when the last day of Pesach falls on Erev Shabbos, is Pirkei Avos recited that first Shabbos?
There are different customs regarding this matter. Practically the widespread custom amongst Anash in Eretz Yisrael is to delay saying Pirkei Avos until the next Shabbos in order to be able to say it together with the Diaspora. [Perhaps however, as a compromise of the opinions, one should say the first chapter the first Shabbos, and then repeat it again the next Shabbos when the Diaspora begins reading it.]
In Eretz Yisrael, when Shavuos falls on Erev Shabbos, is Pirkei Avos recited that following Shabbos?
Yes. In such a case in Eretz Yisrael they will be one chapter ahead of the Diaspora throughout the reading, until Rosh Hashanah.
 Siddur Admur
 There are six Shabbosim between Pesach and Shavuos corresponding to the six chapters of Pirkei Avos.
The reason for saying Pirkeiy Avos between Pesach and Shavuos: As a preparation for Matan Torah.
 Sichas Parshas Naso 1982 [Hisvadyus 3/1633]; So was the custom of the Rebbe Rashab. [Rebbe in Reshimos brought in Otzer p. 244]
The widespread Chabad custom until the year 1978 was to learn Pirkeiy Avos up until Shavuos and not onwards. However from 1978, based on a Sicha of the Rebbe on the Shabbos after Shavuos, the custom became to say Pirkeiy Avos also on the Shabbosim after Shavuos.
 Siddur; Admur in 54/4
The reason: The reason we conclude with these verses is because it is customary to recite Kaddish after Pirkeiy Avos, and we do not recite Kaddish unless we read a teaching that contains a verse or an expounding of a verse. [54/4]
 Otzer Minhagei Chabad p. 246 based on Sichas Netzavim Vayeilech 1987
The Rebbe in Sichas Parshas Tezei 1981 plainly mentioned that the custom is to say it only once. However in later years, the Rebbe publically debated whether it should be said once or twice. [See Hisvadyus 1985 5/2887; Hisvadyus 1987 4/352; Hisvadyus 1989 4/279; Sefer Hasichos 1990 2/681; Sefer Hasichos 1991 2/797] This led to a variety of opinions on this matter.
 Hiskashrus 946 based on Sichas Parshas Tezei 1981
 Siddur Tehilas Hashem Kehos Eretz Yisrael based on Hisvadyus 1990 4 page 259 footnote 104 and Hisvadyus 1987 4 page 400
 Sichas 1991 2 p. 597
 It is the Ashkenazi custom to read the first chapter of Pirkei Avos on this Shabbos. [Luach Eitz Chaim; Bein Pesach Leshavuos p. 162; Luach Dvar Beito] The Sefardi custom is to begin it only the next Shabbos.
The Chabad Custom:
In Sichas Shabbos Parshas Bamidbar 1989 [Sefer Hasichos p. 485 footnote 66] the Rebbe mentions that there are usually six Shabbasim between Pesach and Shavuos corresponding to the six chapters in Pirkeiy Avos. However in Eretz Yisrael there are at times seven Shabbosim. The Rebbe then says there are different customs regarding the seventh Shabbos which is directly prior to Shavuos. Some repeat the sixth chapter again while others begin again from the first chapter. This implies that according to all customs they would already begin the reading on the first Shabbos. The Rebbe then concludes that when there is a dispute between the Diaspora and Eretz Yisrael the ruling is like the Diaspora. In Sichas Naso 1982 3/1633 the Rebbe says that in the Diaspora they are learning the first chapter while in Eretz Yisrael they are already learning the second chapter, hence giving credence to the difference of order between Eretz Yisrael and the Diaspora. Practically, in Otzer Minhagei Chabad p. 247 they write, based on the above Sicha, that in Eretz Yisrael they are to study the first chapter that Shabbos. However Luach Kolel Chabad writes that in Eretz Yisrael one is not to begin Pirkeiy Avos until the next Shabbos, and so concludes Hiskashrus 45. Their reasoning is in order to avoid making a differentiation between their reading and the reading in the Diaspora.
 So concludes Luach Kolel Chabad; Hiskashrus 45 that is the proper custom to follow.
 As seems to be the Rebbe’s opinion as based on the above Sichas.
 Sichas Naso 1982 3/1633; Luach Kolel Chabad; Hiskashrus