László Felkai started playing water polo as a young boy in 1950 with Ferencvárosi TC and played there until 1970. With Ferencvárosi, Felkai won Hungarian titles in 1962, 1963, 1965 and 1968, while taking silvers in 1961, 1964, 1966 and 1969 and bronzes in 1967 and 1970. He also won the Hungarian Cup in 1957, 1962, 1964, 1965, 1967 and 1969. Felkai finished his sporting career by playing with Budapesti Spartacus from 1970-77. He was capped 120 times with the Hungarian national team between 1960-74 and won an Olympic gold in 1964, Olympic bronzes in 1960 and 1968, a European title in 1962, and Universiade titles in 1963 and 1965, while taking silver at the 1959 Universiade and bronze at the 1961 Universiade. He was elected as the best Hungarian water polo player of the year in 1964 and 1968. After finishing his sporting career, Felkai worked as a water polo coach, coaching such Hungarian teams as Budapesti Spartacus, Ferencvárosi TC, KSI SE, Budapesti VSC, Vasas SC, Tungsram SC and Budapesti Honvéd SE.