Feller was born in Zagreb to Ida Oemichen-Perc, a Croatian-Austrian Catholic, and Eugen Victor Feller, who was born to a Polish Jew named David Feller and an Austrian Catholic named Elsa Holzer. Eugen was a famous chemist and created Elsa fluid named after his mother. According to Gian-Carlo Rota, Feller's father's surname was a "Slavic tongue twister", which William changed at the age of twenty — but as can be seen, this claim was false. His christened name, Vilibald, was chosen by his Catholic mother for the saint day of his birthday. In his school documentation, the small municipality of Donja Stubica in Zagorje is mentioned. This is the birthplace of his father, who was an apothecary and owner of a company producing hygienic utensils and cosmetics.

William finished his elementary and middle education in Zagreb, as well as two years of his math study. From 1925, he continued his study in Göttingen, Germany, where he gained the doctoral degree in 1926 under the supervision of Richard Courant, with his work Über algebraisch rektifizierbare transzendente Kurven.

Feller held a docent position at the University of Kiel beginning in 1928. Because he refused to sign a Nazi oath, he fled the Nazis and went to Copenhagen, Denmark, in 1933. He also lectured in Sweden (Stockholm and Lund). Finally, in 1939 he arrived in the U.S. where he became a citizen in 1944 and was on the faculty at Brown and Cornell. In 1950 he became a professor at Princeton University. The works of Feller are contained in 104 papers and two books on a variety of topics such as mathematical analysis, theory of measurement, functional analysis, geometry, and differential equations.

He was among the foremost probabilists outside of Russia. In the middle of the 20th century, probability was not generally viewed as a fruitful area of research in mathematics except in Russia, where Kolmogorov and others were influential. Feller contributed to the study of the relationship between Markov chains and differential equations. He wrote a two-volume treatise on probability that has since been universally regarded as one of the most important treatments of that subject.

Numerous topics relating to probability are named after him, including Feller processes (those possessing the Feller property), Feller's explosion test, Feller-Brown movement, and the Lindberg-Feller theorem. Books written by him and published as textbooks are considered invaluable in the popularisation of the theory of probability and among the best written during the 20th century. Feller made fundamental contributions to, among other things, renewal theory, Tauberian theorems, random walks, diffusion processes, and the law of the iterated logarithm.

Despite the fact that he spent the better part of his life out of Croatia where he was born and grew up, and where he started his education, he was in touch with his relatives there, and with his colleagues at University of Zagreb, whom he sometimes visited and lectured. He received numerous awards and was an honoured member of numerous educational institutions (in Boston, Zagreb, London, Copenhagen).

Feller was among those who initiated the publication of the now well-known review journal Mathematical Reviews.