After finishing primary school, which he attended in Nova Cerkev and Dobrna and graduation a classical gymnasium #2 in Maribor in 1974, he joined a religious order of the Society of Jesus and after the novitiate, that he mast interrupt because of compulsory military service in the Yugoslav Army in 1976, Alojzij consequently studied at the Theological faculty at the University of Ljubljana from 1977 and continued his studies at the Pontifical Gregorian University in Rome, Italy from 1980.

[1] He professed as a Jesuit and was ordained as priest on 3 July 1983 by bishop Jožef Smej,[2] after completed his philosophical and theological studies.

Cvikl served as a chaplain in the parish of Ljubljana-Dravlje and after that continued his postgraduate studies at the Lumen Vitae Faculty of the Saint-Louis University in Brussels, Belgium from 1988 until 1990 with a master's degree of the pedagogy.

After that he served as the Rector of the Pontifical Collegium Russicum in Rome (2001–2010), but in 2010 returned to Slovenia and worked as an economist of the Archdiocese of Maribor.

On 26 April 2015, he was consecrated as bishop by Archbishop Juliusz Janusz and other prelates of the Roman Catholic Church in the Cathedral of Saint John the Baptist in Maribor.