Born in Mezdra, Genkov began his career in the local Lokomotiv, previously having been a track and field athlete. He made his debut during the 2001–02 season on 14 October 2001 in a 0–2 away loss against Sitomir Nikopol in the campaign of V AFG. On 12 May 2002, Genkov scored his first goal in a 2–2 home draw against Olimpik Teteven. As of the 2002–03 season he became first-choice forward scoring 14 goals. After 24 goals in the next season he transferred to Lokomotiv Sofia.
He made his debut for Lokomotiv Sofia on 6 August 2004, in a 0–2 away loss against Litex Lovech. In Sofia, Genkov established himself in one of the leading A PFG clubs and helped his team to reach the fourth place in 2005–06 and secure a place in the 2006–07 UEFA Cup. During the qualification rounds, Genkov scored four goals in six matches. In Bulgaria's A PFG, he became the top goalscorer of 2006–07 with 27 goals, leading Lokomotiv to the third position and 2007–08 UEFA Cup qualification.
On 9 June 2007, he signed a four-and-a-half-year contract with Dynamo Moscow for a transfer fee of €2.25 million.
Return to Lokomotiv Sofia
In January 2010, Genkov was loaned out to his former club. On 7 March 2010, he netted twice in the 5–1 away win against FC Sportist Svoge. In the following season, with 11 goals to his name, Genkov had climbed up to second place in the scoring charts in the A PFG by the time his loan expired.
On 28 January 2011, Genkov joined Polish Ekstraklasa side Wisła Kraków on a three and a half-year deal for an undisclosed fee from Dynamo Moscow. He won the Ekstraklasa championship in his debut season. Genkov was club's top goalscorer in the second part of the season with six goals in the league. In the following season, he was a starter in all six matches in the Wisła's ultimately unsuccessful UEFA Champions League qualifying campaign. On 9 September, Genkov picked up an injury at beginning of the league match against Lech Poznań. He returned to play on 15 October, scoring the winning goal in a 3–1 home win over Jagiellonia Białystok. Genkov also netted the winning goal in a 2–1 victory over FC Twente, in the last UEFA Europa Leaguegroup stage match, which lead his team to qualify for the round of 32. On 16 February 2012, he scored a late equaliser in the first leg tie against Standard Liège. Wisła Kraków played with ten-men for most of the match after Michał Czekaj was sent off in the 27th minute. He scored his first hat-trick for Wisła Kraków on 14 April, scoring all three of his team's goals in a 3–2 home win over ŁKS Łódź in the Ekstraklasa. Genkov was a top goalscorer of 2011–12 Polish Cup with four goals from three games.
Genkov signed with Levski Sofia on 19 June 2013 on a three-year deal. He was not registered for the 1st round of the UEFA Europa League games against Kazakh club Irtysh Pavlodar, which his team lost by an aggregate score of 0–2. Genkov made his official debut for Levski Sofia on 21 July 2013, in the 1–2 away loss against Botev Plovdiv in an A PFG match. He scored his first goal for the team from Sofia a week later in the 1–2 home defeat inflicted by Lokomotiv Plovdiv.
