This summary is drawn from the Rebbe's letters to him (about 5714–5726, 1954–1966); I did not find a published biography.
Torah and halachic questions. Much of the correspondence consists of the Rebbe's answers to scholarly questions: textual issues in Targum Yonasan and the recitation of related passages (IK 4594), mentioning the Korban Pesach in Selichos and reciting the verses of Mussaf (IK 4743), and whether there is concern about the names Yehuda or Shmuel because of Rabbi Yehuda HaChasid's instructions — the Rebbe wrote there is not (IK 2890). In Chassidus he explained causality in creation and why only in the lower worlds, especially Asiyah, does a created thing perceive itself as separate (IK 7601).
Study and customs. The Rebbe advised on the order of studying Shnayim Mikra and Chassidus when learning alone (IK 7368), encouraged celebrating birthdays with chassidic customs (IK 6927), and wrote about celebrating a bar mitzvah on its actual date — even when it falls on Pesach — with joy (IK 7201, 7224).
Publishing. On sending a set of the Sdei Chemed and the costs of printing, the Rebbe stressed how much Torah publishing depends on communal support (IK 7948). For Shavuos, he blessed him to receive the Torah with joy and inner meaning (IK 9327).
Sources: the Rebbe's letters cited above.