He started playing tennis at the age of six years. His father is Chander and mother is Indu. He is the youngest in the family. His sisters are Ankita Bhambri and Sanaa Bhambri, and he is the cousin of Prerna Bhambri and Prateek Bhambri, all of whom are professional tennis players. His tennis coach is Aditya Sachdeva.
Bhambri reached the junior No. 1 ranking in the world in February 2009. He won the Junior Australian Open Boys singles title in 2009 by beating Alexandros-Ferdinandos Georgoudas of Greece in the finals.
He competed in all the junior Grand Slams in 2008. He lost at the 2008 Australian Open Boys' Singles in the semifinals to eventual champion Bernard Tomic.
Yuki concluded his season by making history as the first Indian winner of the prestigious Orange Bowl at age 16, beating Jarmere Jenkins of the U.S. in straight sets.
In 2009, Bhambri began the year by winning the junior Australian Open, defeating Greek Alexandros-Ferdinandos Georgoudas in the final in only 57 minutes. Throughout the tournament, he only dropped one set, becoming the first Indian singles winner of the junior Australian Open and the fourth Indian in history to capture a junior singles grand slam title.
After his win in the Orange Bowl, Yuki received a spot in the main draw of the Sony Ericsson Open. He also obtained a wild card spot in the doubles main draw pairing with Harsh Mankad as well as a wild card spot in the singles qualifying draw in the Chennai Open, where he lost to top college player and compatriot Sanam Singh in three sets. Yuki also obtained a wild card position for the 2010 Australian Open qualifying draw as an additional reward for his win at the 2009 Junior Australian Open. At the 2009 Sony Ericsson Open, Yuki lost to Diego Junqueira in the first round. Following his showing at the Sony Ericsson, Yuki played in an ITF Futures Event in New Delhi, India, where he won the tournament becoming the youngest Indian to win a Futures event; Yuki continued this winning streak by capturing the title of the following Futures event held in the same venue. Yuki made his Davis Cup debut against South Africa in the World Group Play Offs.
In August, Bhambri advanced to the finals of the first Youth Olympics Games in Singapore by defeating Damir Džumhur in three sets. He played Colombian Juan Sebastian Gomez in the finals. After suffering cramp in his thigh, he conceded the match with the score at 1–4 in the third set. This gave him a silver medal at the first Youth Olympics in Singapore.
Yuki Bhambri defeated Amir Weintraub in straight sets to win the first ever ATP Challenger singles title of his career in Fergana, Uzbekistan on 20 May 2012.
He entered his first ATP Challenger final of the season at Kaohsiung, Taiwan. He lost to Chinese Taipei's Lu Yen-hsun in the final.
He won his first ATP Challenger singles title of the season and second of his career at Traralgon, Australia. He defeated American Bradley Klahn in the final. This was his second ATP Challenger singles title of his career.
Bhambri entered the 2014 Australian Open in both singles and doubles. He lost to Potito Starace in first round of the singles qualifying draw. He entered the men's doubles with Michael Venus from New Zealand as his partner. They defeated Roberto Bautista Agut and Daniel Gimeno-Traver in the first round. In the second round, the pair shocked tenth-seeded Jean-Julien Rojer and Horia Tecau by defeating them in straight sets. They finally lost to 5th seeded Indo-Czech pair of Leander Paes and Radek Stepanek in round 3.
On 8 Feb 2014, Bhambri won his third ATP Challenger singles title with a win at Sriram Capital PL Reddy Memorial ATP Challenger tennis tournament in Chennai. He defeated Alexander Kudryavtsev of Russia in three sets in the final. He also won the doubles title at same event with his partner Michael Venus of New Zealand. The pair defeated Indo-Slovakian combination of Sriram Balaji and Blaz Rola in the doubles final.
Later in the year, Bhambri won two medals for India at the 2014 Asian Games. He won bronze in the men's singles category. He lost to Japan's Yoshihito Nishioka in the semifinals. He also won bronze in men's double category with Divij Sharan as his partner. The pair lost in the semifinals to Korean pair of Lim Yong-kyu and Chung Hyeon.
Starting the year as 315th ranked player in the ATP rankings, Bhambri qualified for the 2015 Australian Open, but lost to Great Britain's Andy Murray in the first round in straight sets.
He then entered 2015 Delhi Open as a Wildcard and reached the finals, losing to Somdev Devvarman in three sets 3-6 6-4 6-0. He maintained his good start of the season and reached the semifinals of Kazan Kremlin Cup. On 13 September, he defeated Wu Di in the finals of Shanghai Challenger in three sets 3-6 6-0 7-6 to win his first title of the season, which helped him reach no. 125 in the ATP Rankings. He reached the finals at the OEC Kaohsiung, losing to Hyeon Chung in straight sets 7-5 6-4.
He broke into the top 100 for the first time in his career on 19 October 2015. He won his first title in India on 31 October defeating Evgeny Donskoy in straight sets 6-2 7-6 in Pune. As a result of his good run, he rose to career best ranking of 88 on 9 November. He finished year with year end ranking of 93.
Bhambri entered main draw of Australian Open for second consecutive year. He lost in the first round to the sixth seed Tomas Berdych in straight sets.
In February he won his sixth doubles challenger title at 2016 Delhi Open partnering with Mahesh Bhupathi.
Bhambri was out of the action from March to September because of tennis elbow injury. This forced his rankings to drop out of 500.
Bhambri started his 2017 campaign with Chennai Open. He won qualifying and entered main draw. He won first round by defeating compatriot Ramkumar Ramanathan but lost in round 2 to Benoit Paire. He then entered qualifying draw of Australian Open where he lost in final qualifying round to Ernesto Escobedo.
In July, Bhambri entered his first quarterfinal at ATP 500 event at the Citi Open in Washington, D.C.. This was his second quarterfinal at ATP world tour. Bhambri defeated Stefan Kozlov in the first round and shocked 6th seed and defending champion Gaël Monfils in the second round. He then defeated Guido Pella to reach the quarterfinals where he lost to eventual finalist Kevin Anderson.