He began his artistic education between 1890 and 1891 with Wojciech Gerson's drawing class in Warsaw. Between 1891 and 1893 he attended the School of Fine Arts in Kraków, and learned in the studios of Ignacy Jabłoński, Władysław Łuszczkiewicz, Florian Cynk, and Józef Unierzyski. In 1893 he continued his studies in Munich in the private schools of Stanisław Grocholski and Anton Ažbé.
In 1894 he left for Paris, where he studied at the Académie Julian in the studios of B. Constant, R. Collin and J.-P. Laurens. In 1897 he returned to Munich to improve his painting skills at S. Hollósy’s school, and he also lived in Hungary, where Hollósy founded an artists’ colony. In 1898 he settled in Italy, which became his artistic homeland for over 20 years; he lived in Rome, travelled to Anticoli Corrado, Sorrento, Amalfi and Capri, and visited Venice, Padua, Ravenna, Florence and Siena. He participated in the life of the Polish artists' colony in Rome.
Between 1901 and 1907 he worked as an illustrator with the elite Warsaw-based artistic and literary magazine Chimera. Starting in 1906, he painted drawings for the Munich magazine Jugend. He also created illustrations for books, including Kasprowicz’s Miłość (Love, Lviv, 1902), Leopold Staff’s Mistrz Twardowski: Pięć Śpiewów o Czynie (Master Twardowski: Five Songs about the Deed, Lviv, 1902) and Stefan Żeromski's Duma o Hetmanie (Elegy for a Hetman, Warsaw, 1909). From 1908 Okuń belonged to the Odłam (Fraction) group. In 1921 he settled again in Warsaw, and travelled to Italy and southern Europe, for example, to Dubrovnik. From 1930 he was a lecturer at the W. Gerson School of Fine Arts in Warsaw.
His works were exhibited regularly at the Zachęta Society of Fine Arts (1893, 1895, 1896-1899, 1901, 1903, 1905, 1907-1916, 1922-1926, 1928-1932, 1934-1936, 1937-1938), at the Krywult Salon (1899, 1900, 1902, 1903, 1904-1905), the Kulikowski Salon (1906) and the Abe Gutnajer Salon (1915, 1917). The artist presented his works also at the Society of Friends of Fine Arts in Kraków (1898-1901, 1902, 1904-1905) and Lviv (1901, 1910, 1911/1912, 1922). He also presented his works in Łódź (1912, 1922), Poznań (1911), Toruń (1927) and Vilnius (1932). A number of individual exhibitions of Okuń's works took place in the the Zachęta Society of Fine Arts (1902, 1906, 1911, 1913, 1919, 1920, 1924, 1929, 1933, 1939). The artist participated in foreign exhibitions in Paris (1895, 1921), Berlin (1899), Kiev (1900), Munich (1906-1907, 1928) and Rome (1916-1917).
The thematic repertoire of Okuń’s paintings includes landscapes, portraits and fantastic compositions. His paintings were characterised by sophisticated decor and rich painterly substance. In the early phases of his work he used a flexible Art Nouveau line and a rich array of light, qualitatively modulated colours such as in Pejzaż z Anticoli (Landscape of Anticoli, 1906). In Italian landscapes, he masterfully created the illusion of spatial depth, evocatively rendered luministic effects, perfectly reproduced the trembling of heated air and saturated images with an inner brightness. A pleasant atmosphere was sometimes complemented by staffage introducing the narrative element, while the outlines additively piled up, subjected to the synthesising geometrisation of the buildings in Renaissance towns, connected the viewer’s imagination with the distant past (Autoportret z Żoną na Tle Anticoli-Corrado / Self-Portrait with Wife against Anticoli-Corrado, 1900).