Found some more background:
Daily Recitation of Tehillim
In the midst of a perilous period for Chabad and indeed, for much of Jewry. The Soviet establishment launched an oppressive campaign in all the territories under its vast dominion with the design of suffocating every last vestige of the Jewish soul – Torah education, belief in G-d, and the practical observance of the mitzvos. Despite the far reach of their secret police and the sheer terror of their ruthless methods, the Communists discovered that a lone citizen was successful in consistently and methodically unwinding their evil designs on a massive scale.
That man was Rabbi Yosef Yitzchok Schneersohn, who formed and personally directed a breathtaking labyrinth of underground Torah schools, synagogues, mikvehs, and the like. He sent overt and covert emissaries, teachers and instructors who were willing to risk their lives to keep the flame of Torah and mitzvos alight in every region of the USSR. He also founded branches of his Torah schools outside the USSR, in Poland, Uzbekistan and in America.
Rabbi Yosef Yitzchok was hounded mercilessly by the Soviets, but he ignored their intimidation. In 5687 (1926), however, he sensed that the enemy was poised to strike a deadly blow. In response, he requested that the book of Tehillim as it is divided into daily portions to be completed in the course of a month, be recited in all synagogues.
Listen, Chassidim, and all Jews who await the coming of Moshiach! Repeat in my name to all Chassidim in the world that I have instructed them to recite a portion of Tehillim as divided by the days of the month in all Chassidic synagogues after morning prayers each day, including Shabbos.
Excerpt of 1927 handwritten directive discovered on his desk on the day of his arrest
Rabbi Yosef Yitzchok was arrested and suffered a brutal incarceration. The Soviets swiftly sentenced the Rebbe to death – a decree that was miraculously reversed, with the help of enormous international outcry. So complete was the reversal that the Soviets were subsequently compelled to escort Rabbi Yosef Yitzchok with his family and his possessions out of the USSR’s iron borders to freedom.
An account by the Rebbe’s close confidant and devoted follower, Rabbi Eliyahu Chaim Althaus, immortalizes Rabbi Yosef Yitzchok’s reflection on the effectiveness of reciting the daily Tehillim:
These were his exact words: “Before 5687 (1927) I was greatly afraid. I did not speculate what my own fate would be, for I never imagined what I would have to endure. Rather, my fear was for the Chassidim. Before I instructed them to begin saying Tehillim, it was really difficult for me.”
“(With regard to) the recital of Tehillim (daily, after davening) – there is no distinction (i.e. it applies equally) between Shuls of ‘Anash’ (a reference to the Chabad Chassidic Fellowship) and to those who follow the Ashkenazic or Polish nusach – may G-d be with them. On account of Ahavat Yisrael, love of our fellow…and especially in view of the importance and inner purpose of saying Tehillim with a minyan, which literally affects all of Israel, materially in “children, life and sustenance,” and spiritually …we must endeavor with every means possible that it become an established practice in every shul of every nusach.”
– From a letter of the Previous Rebbe, Rabbi Yosef Yitzchok
Tehillim Corresponding to One’s Age
“(It is well known the custom which the Alter Rebbe received)…from his master, who in turn received it from his master the Ba’al Shem Tov, to recite the chapter of Tehillim corresponding to one’s age, after the morning prayer (Shachris), prior to reciting the customary daily Tehillim…”
– From ‘Kovetz Michtavim’ – a selection of letters by Rabbi Yosef Yitzchok Schneerson, regarding Saying Tehillim