Chung took up tennis as a way to try to help maintain his eyesight after requiring glasses at a young age. He won the Eddie Herr International and Junior Orange Bowl Boys under-12s titles in December 2008, and was subsequently signed, along with his brother Chung Hong, to the Nick Bollettieri Tennis Academy at IMG in Florida. He began competing on the ITF junior tour in 2012, and was runner-up in the 2013 Wimbledon Boys' Singles, a month after winning his first Futures title. He later competed in his first ATP tournament, the Malaysian Open, being defeated in the first round. He reached a career junior high of number 7, with an 84-32 win-loss record.
2014 saw Chung move full-time to the men's professional game, winning 3 Futures tournaments and the 2014 Bangkok Open, his first Challenger level tournament. He competed in the qualifying for the 2014 US Open and won two matches for the South Korea Davis Cup team to help keep them in the Asia/Oceania Zone Group I. He also won gold in the doubles competition at the 2014 Asian Games and ended 2014 ranked 151 in the ATP Rankings.
Chung reached the final qualifying round for the Australian Open, but focused his efforts on the Challenger Tour. He won at the Burnie Challenger in February to reach the world's top 150 and subsequently received a Wildcard for the ATP World Tour competition at the Miami Open, getting to the second round of this Masters level tournament. Two further Challenger titles followed in April and May 2015, which saw him enter the world top 100 for the first time. His rapid rise up the rankings and an error on behalf of the Korean Tennis Federation meant that he missed the entry deadline for the 2015 French Open. Although he was later handed a wildcard into the qualifying tournament, he was eliminated in the first round. Chung then lost in straight sets to unranked Nicholas Monroe in the first round of qualifiers in the Topshelf Open. In the 2015 Wimbledon Championships, he reached his first main draw in a Grand Slam tournament, losing 10–8 in the decisive set in the first round to Pierre-Hugues Herbert. At the 2015 US Open, Chung qualified for the main draw and recorded his first win in a Grand Slam against James Duckworth. Chung lost in the second round against fifth seed Stanislas Wawrinka in straight sets, despite taking each set to a tiebreak. He continued to play on both the Challenger and ATP tour through 2015, winning a further Challenger at Kaohsiung in September and reaching his first ATP quarter-final at the Shenzhen Open. He was awarded the year-end ATP Most Improved Player award for 2015, after climbing over 120 places to 51st in the rankings.
Chung won his first round match in Brisbane against Sam Groth. He lost in the second round to the 3rd seed Marin Čilić. At the Australian Open Chung lost in the first round to Novak Djokovic.
In February, Chung lost in Sofia in the first round to qualifier Marius Copil. In Rotterdam Chung reached second round, where he lost to Viktor Troicki. In Marseille he lost in the first round to David Goffin. In Dubai Chung defeated Andreas Seppi in the first round. Then he lost to Roberto Bautista Agut, winning only a single game in this match.
Chung won both singles in the Davis Cup tie against New Zealand. South Korea won 3–1.
In Indian Wells, Chung lost in the first round to Albert Ramos-Viñolas. In Miami he lost in the first round to Denis Kudla.
Chung scored his first Australian Open match by defeating Renzo Olivo. He lost in the second round against eventual semifinalist Grigor Dimitrov despite winning the first set. In April, Chung reached the quarter-finals of the 2017 Barcelona Open Banco Sabadell, taking Rafael Nadal to a tie-break before eventually losing 6-7 2-6. To reach the quarter-finals, Chung came through qualifying and the main draw to record five wins in a row without dropping a set, including victories against Denis Istomin, Phillip Kohlschreiber and Alexander Zverev.
In May, Chung competed at the 2017 BMW Open in Munich, reaching the semi-finals. In the second round, Chung recorded a 6-2 6-4 victory against Gael Monfils, who at a ranking of 16 was the highest-ranked player Chung had defeated. At the French Open, Chung achieved his best Grand Slam result by reaching the third round, including a win over the No. 27 seed Sam Querrey in the first round. He lost his all-Asian encounter with the No. 8 seed Kei Nishikori in 5 sets.
In August, Chung reached the third round of the Rogers Cup, his best result at an ATP Masters 1000 tournament to date, beating 13th-ranked David Goffin in the 2nd round in straight sets.
Updated through the 2017 US Open.