His family was evacuated to Tula during World War II, and his father died in the Battle of Kursk.

Although scoring respectably in his first two attempts at qualifying for the Soviet Final (10/18 at Kharkov 1956 and 10½/19 at Kiev 1957), he failed to advance, but was gathering experience and strength at high levels.

At the Moscow International of 1959 (Alexander Alekhine memorial), Vasiukov tied for 4th-6th places with 6/11, behind only winners Vasily Smyslov, Boris Spassky, and Bronstein.

Vasiukov proved his grandmastership at East Berlin 1962 with a powerful victory, at 11½/15, ahead of Leonid Stein.

During his peak years, from the early 1960s to the mid-1970s, Vasiukov scored wins in individual games over many top Soviet players, such as Smyslov, Bronstein, Tigran Petrosian, Mikhail Tal, Paul Keres, Mark Taimanov, Efim Geller, and Lev Polugaevsky.

He was unable to defeat top-ranking Soviet stars such as Viktor Korchnoi, Anatoly Karpov, Boris Spassky, or Stein.

The Soviet Championships were usually the strongest tournaments in the world during Vasiukov's main competitive period, surpassing the strength of Interzonals, with very few if any weaker opponents, since the arduous qualifying process eliminated the outsiders.

The next year at Kharkov (URS-ch35), the format was a Swiss system, and Vasiukov was just above the middle of the pack, as Tal and Polugaevsky won.

In the 1975 Vilnius Zonal, he scored 7½/15 for tenth place, as the winners were Boris Gulko, Vladimir Savon, Yuri Balashov, and Vitaly Tseshkovsky.

Since Vasiukov never placed near the top in Soviet Zonal competition, he was unable to earn an opportunity to play in an Interzonal tournament.

Vasiukov was unable to maintain his peak of the early 1960s, but he remained remarkably competitive for many years afterwards, especially in international tournaments outside the Soviet Union which had mixed-strength fields, where he scored most of his successes.

Vasiukov had the top performance of his career with his win at Manila 1974 at 10½/14, as he was ahead of Petrosian and Bent Larsen, among others, in a top-class field.

He played five times in the traditional match against Yugoslavia, at Lvov 1962, Rijeka 1963, Sukhumi 1966, Skopje 1969, and Tbilisi 1973.