I did not find a reliable published biography of Eliyahu Amikam; this summary is drawn from the Rebbe's letters to him (late 1960s–early 1970s), which show him to be a writer in Israel who used his pen on Jewish issues.
The power of the pen. The Rebbe encouraged him to use his talents for Torah truth without compromise, influencing others through his writing (IK 10480), and wrote that words spoken sincerely, including words shared from a farbrengen, deeply affect others (IK 9385).
Jewish unity and identity. On proposals to renew the Sanhedrin, the Rebbe wrote that Jewish unity and spiritual health depend on the daily observance of Torah and mitzvos by every Jew (IK 9569). He stressed the seriousness of the "Who is a Jew" question, explaining that such pressing communal issues had delayed his correspondence (IK 9890). These letters present the Rebbe's own position on public questions.
Every Jew's potential. The Rebbe taught him that every Jew has infinite potential rooted in his Divine soul, so that even one person can bring about great change, with quality mattering more than quantity (IK 8611), and encouraged practical steps toward positive goals, especially at Chanukah (IK 9794).
Sources: the Rebbe's letters cited above.