Tarkowski grew up in New Moston, Manchester and joined League One side Oldham Athletic on a two-year scholarship in May 2009. He made his reserve team debut in October 2009. A centre back, Tarkowski was called up to the first team for the first time for a league match versus Plymouth Argyle on 15 January 2011. His senior debut came on 22 January, coming on as an 80th-minute substitute for the injured Neal Trotman in a 2–1 home win over Brentford. Tarkowski made his first Latics start in a 1–0 defeat to Leyton Orient on 12 March and played the full 90 minutes. He was awarded a two-year professional contract by the club in May 2011. He made 9 appearances during the second half of the 2010–11 season.
Tarkowski had to wait until 10 December 2011 to make his first appearance of the 2011–12 season, when he came on as a 61st-minute substitute for Jean-Yves Mvoto in a 2–0 home defeat to Sheffield Wednesday. A half time substitute appearance in Oldham's 2–1 Football League Trophy Northern Area final first leg defeat to Chesterfield on 30 January 2012 kicked off a run which saw Tarkowski included in almost all of Oldham's first team squads until the end of the season. Tarkowski crowned his run of appearances by scoring his first senior goal for the club on the final day of the season in a 2–1 win over Carlisle United. He made 18 appearances during the 2011–12 season and scored one goal.
Tarkowski began the 2012–13 season as a first choice centre back alongside Jean-Yves Mvoto and started Oldham's first six games of the season. He scored his first goal of the season in a 2–0 league win over Crewe Alexandra on 2 October 2012. Tarkowski played a part in Oldham's run to the fifth round of the FA Cup, playing in the first round win over Conference Premier side Kidderminster Harriers and starting in the home draw and away replay defeat to Premier League side Everton in the fifth round. He scored his second goal of the season in a 2–1 league win over Stevenage on 19 February 2013. Tarkowski made 26 appearances and scored two goals during the 2012–13 season. He signed a new two-year contract in the summer of 2013.
Tarkowski began the 2013–14 season with a bang, scoring Oldham's first goal in a 4–3 away league win over Stevenage on the opening day. He was picked by the Latics supporters as Oldham's Player Of The Month for August 2013. Now lining up at centre back alongside Jonathan Grounds, Tarkowski scored his second goal in a 2–1 defeat to Preston North End on 9 September 2013. Tarkowski was an ever-present in Oldham's runs to the Northern Area semi-final of the Football League Trophy and the third round of the FA Cup. He started the FA Cup third round tie against Premier League side Liverpool at Anfield and scored an 82nd minute own goal, which consigned the Latics to a 2–0 defeat. Tarkowski's form saw him offered a new contract extending until the end of the 2016–17 season on 22 January 2014, but he departed Boundary Park 9 days later. During his time with the Latics, Tarkowski made 89 appearances and scored six goals.
Tarkowski signed for League One side Brentford on a three-and-a-half-year deal for an undisclosed fee on 31 January 2014. After a period on the bench, he made his Brentford debut with a start in a 0–0 draw away to Carlisle United on 1 March. He scored his first goal for the club on his third appearance, heading in to send the Bees on their way to a 2–0 victory over Tranmere Rovers on 11 March. Four days later, Tarkowski was sent off for the first time in his career during a 1–0 win over Leyton Orient. Following Brentford's promotion to the Championship on 18 April, Tarkowski scored his second goal for the club in a 2–2 draw at Milton Keynes Dons on 21 April. He finished the 2013–14 season with 13 appearances and two goals.
Tarkowski began his first season at Championship level as an ever-present in the Bees' league matches. He suffered a dip in form in September and early October, stepping up to take his first penalty for the club in a league match versus Leeds United on 27 September and hammering the ball over the bar. He was spared his blushes, with Jota and Alan McCormack on target in the eventual 2–0 win. Tarkowski gave away a penalty in a 2–1 defeat to Watford on 30 September and received his fourth yellow card of the season (his third in four matches). Another yellow card in a 0–0 draw with Wigan Athletic on 18 October saw him incur a suspension and he failed to appear again until 28 December, scoring what proved to be Wolverhampton Wanderers' winner with an own goal in a 2–1 defeat. Despite the blunder, Tarkowski held onto his starting place and scored his first goal of the season against Reading on 25 April 2015, making sure of the points with the second in a 2–0 win. Tarkowski had the opportunity to convert a late penalty to make it two goals in two games in the following match against Wigan Athletic, but his miss made little difference with the Bees already 3–0 ahead. He appeared in both legs of Brentford's 5–1 playoff semi-final aggregate defeat to Middlesbrough and finished the 2014–15 season with 38 appearances and one goal.
Tarkowski helped salvage a 2–2 draw with Ipswich Town on the opening day of the 2015–16 season, feeding Andre Gray to score in injury time, before bundling home a 96th-minute equaliser to earn a place in the Football League Team of the Week. He suffered a displaced fracture of the nose in the following match versus Bristol City, but kept his place alongside Harlee Dean until suffering a calf injury in a 1–1 draw with Leeds United on 12 September. He missed two matches before being sent off on his return in a 2–1 defeat to Sheffield Wednesday. Amidst a number of transfer bids, Tarkowski was internally disciplined for refusing to play in Brentford's match versus Burnley on 15 January 2016, for which he later apologised. He failed to appear for the Bees again and departed Griffin Park two weeks later, having made 74 appearances and scored 4 goals in two years with the club.
On 1 February 2016, Tarkowski signed a three-and-a-half year deal with Championship club Burnley for an undisclosed fee.
Tarkowski's paternal grandfather left Poland for England after World War II, and married a woman of Irish descent. Tarkowski is therefore eligible for both the English and Polish national teams. Tarkowski attended New Moston Primary School and Failsworth High School, which he left in 2009. He is a Manchester United fan and admired David Beckham and Paul Scholes while growing up.As of match played 23 September 2017.
BrentfordFootball League One runner-up: 2013–14
Oldham Athletic Player of the Month: August 2013