He learned to play chess as a pre-teen and progressed up the ranks to Master by the age of 24, but then his progress pretty much stopped, mostly due to his habit of playing blitz chess. Not blitz as we think of it, but blitz in serious games; he typically only spent 15-20 minutes per game which resulted in careless play.
His first success came in 1955 and 1956 when he won the Soviet Army Championship. In 1958 he failed to qualify for the Ukrainian Championship and was ready to quit chess, but when one of the players who did qualify couldn't play, Stein was given his place. He finished third and exceeded the Master norm by one and a half points and his career took off.
He won the Ukrainian championship, a bronze medal in the 1961 USSR Championship and qualified for the 1962 Stockholm Interzonal where he tied for 6th-8th places with Benko and Gligorich and then won the match for the last candidate spot. Stein won scored 3 points, Benko 2 pts. and Gligoric zero. Unfortunately Stein wasn't allowed to play in the Candidates' tournament at Curacao due to the FIDE rule limiting the number of participants from the same country to three and so Pal Benko went instead.
His luck was no better at the next Interzonal tournament in Amsterdam 1964. He finished in 5th place, but he was again disqualified because of the three player rule; Mikhail Tahl, Vasily Smyslov, and Boris Spassky finished ahead of him. He represented the USSR at the Tel Aviv Olympiad in 1954 and scored 10-3, winning the individual gold medal on the first reserve board. At the Havana Olympiad in 1966 he scored 9-3, winning the silver medal on board four. His play was impressive enough that Bobby Fischer offered to play him in a match immediately after the Havana Olympiad, but nothing came of it.
Again, in 1967, Stein qualified for the Sousse Interzonal and tied for 6th–8th places with Samuel Reshevsky and Vlastimil Hort. The three had a playoff match in Los Angeles which was won by Reshevsky.
Stein again qualified for the Interzonal at Petropolis in 1973 and was considered one of the favorites, but as he was preparing to leave with the Soviet team for the European team championships, he collapsed of an apparent heart attack and died at the age of 38 in the Rossiya Hotel in Moscow.
Stein's style was influenced by Chigorin and Alekhine and he was a highly intuitive, natural player, a brilliant, but sound, attacking genius. But, unlike Tahl, he did not take unnecessary risks. He had even scores against Vasily Smyslov, Tigran Petrosian, and Mikhail Botvinnik and plus records against Mikhail Tal, Boris Spassky, and Paul Keres.
It's hard to pick one of Stein's games because there are so many brilliant ones, but the following miniature against the great Hungarian GM Lajos Portisch from the 1962 Stockholm Interzonal is typical.
There are several good books on Stein available: Leonid Stein: Master of Risk Strategy by Gufeld, Leonid Stein - Master of attack by Raymond Keene and Stein Move by Move by Thomas Engqvist. Any one of them are a good buy if you are looking for great attacking games.