Marta is the daughter of Oleh Kostyuk and his wife, Talina Beiko. Her father was the technical director of the Antey Cup, a junior tennis tournament in Kyiv; her mother was a professional tennis player who reached a career-high WTA ranking of No. 391, and won a $10,000 title in her home city of Kyiv in 1994. Kostyuk started playing tennis at a young age at the Antey Tennis Club, on the west side of Kyiv, coached by her mother. She described her initial experience in tennis at age five: "My mom was always working a lot as a coach, and the first time I went to the courts to train, I just understood that if I started doing tennis, I'd get to spend more time with my mom. So that was kind of my motivation – if I played tennis, I'd be around her more often". She was also coached by her maternal uncle Taras Beiko, who had played for the USSR and Ukraine in the late 1980s and early 1990s.
Kostyuk won the Burnie International, a $60,000 ITF tournament in Australia, in February 2018, and reached the final of the Zhuhai Open, also a $60,000 ITF tournament, in March, but did not sustain her level of success in the rest of the year. In 2019, she won two further ITF titles, and reached the third round of the WTA Strasbourg tournament.
In the US Open, she beat former top 10 player Daria Kasatkina in straight sets in the first round. She then beat former semifinalist Anastasija Sevastova. In the third round she met former champion and world No. 9 Naomi Osaka. She overcame a first set deficit by winning the second in a tie-breaker, but was beaten in the third.
1 The first Premier 5 event of the year has switched back and forth between the Dubai Tennis Championships and the Qatar Total Open since 2009. Dubai was classified as a Premier 5 event from 2009 to 2011 before being succeeded by Doha for the 2012–2014 period. In 2015, Dubai regained its Premier 5 status while Doha was demoted to Premier status. The two tournaments have since alternated status every year.
Marta Kostyk is a daughter of a professional tennis player Talina Beiko, who represented a Ukrainian tennis team. Her uncle Taras Beyko is also a retired tennis player. Marta is a sister of collegiate tennis player Mariya Kostyuk, who competed for Chicago State University. Marta is a cousin of a professional football players Vadym Slavov and Myroslav Slavov.