Campbell’s career began with Saprissa in 2009 before being sent on loan to Puntarenas in January 2011.

Campbell was selected for Costa Rica’s squad for the 2009 CONCACAF U-17 Championship. He was the team’s highest scorer, although he had only two goals in total. That same year, he participated in the 2009 FIFA U-17 World Cup held in Nigeria. He was named in the 2011 CONCACAF U-20 Championship two years later and tied for top scorer with six goals.

On 28 July 2011, Costa Rican television reported that Campbell had come to an agreement with Arsenal, after having been discovered by club scout Danny Karbassiyoon. Campbell later said that while he had not yet signed, talks were taking place. On 6 August, Costa Rican media reported that Campbell was set to sign a five-year deal with Arsenal after rejecting moves to Sevilla and Fiorentina.

On 19 August 2011, Arsenal confirmed that they had signed Campbell, but it was reported eight days later that he had failed to obtain a work permit to enable him to play in England. Campbell finally received that permit in July 2013.

Campbell was first called up to the senior team in March 2011, for friendlies with PR China and Argentina. He would juggle cans on the streets of San José. Before the friendlies, he was spotted doing this by national team manager Ricardo La Volpe. In announcing Campbell’s inclusion in the squad, La Volpe said, “Who is Messi? Who is Neymar? Gentlemen of the press – I present to you: Campbell. I discovered him playing with cans.”

On 6 July 2012, Campbell joined La Liga side Real Betis on a season-long loan. He made his debut on 25 August, coming on as a substitute for Salvador Agra to play the last 20 minutes in their 2–1 home defeat to Rayo Vallecano.

By summer 2013, Campbell had been granted a work permit in England but he joined Greek side Olympiacos on another one-year loan.

On 27 October 2013, Campbell starred in Olympiacos’ win over OFI Crete, picking up four assists to help his side win 5–1. Campbell impressed again on 10 November, scoring once and earning two assists as Olympiacos thrashed PAOK 4–0.

In September 2013, Campbell became the subject of controversy when video emerged of his off-the-ball dive against the United States’ Matt Besler in a World Cup qualifying match. Besler received a yellow card and was suspended. The footage generated discussion of the quality of CONCACAF refereeing, and whether such cheating should be eligible for post-match disciplinary action. Campbell’s dive was described as “pathetic”, “cynical” and “disgraceful”. On 13 September, the Costa Rican media reported that FIFA were investigating Campbell over the incident, although nothing came of the investigation. On 4 October, FIFA officially reprimanded Campbell for unsporting behaviour.

On 25 February 2014, Campbell scored the second goal Olympiacos’ 2–0 win over Manchester United in the last 16 of the UEFA Champions League. Olympiacos, however, did not proceed to the next round after losing 3–0 in the second leg.

Arsenal manager Arsène Wenger announced that Campbell would be part of his first team plans for Arsenal’s 2014–15 season. On 2 August 2014, Campbell scored in Arsenal’s 5–1 win over Benfica in the Emirates Cup. Eight days later in the 2014 FA Community Shield, he replaced Aaron Ramsey for the final five minutes of the 3–0 win over Manchester City at Wembley Stadium. He made his Premier League debut on 23 August in a 2–2 draw against Everton at Goodison Park, coming off the bench to replace Alex Oxlade-Chamberlain on the 74th minute of the match.

On 14 June 2014, Campbell scored Costa Rica’s first goal in the 2014 FIFA World Cup, equalising against Uruguay in an eventual 3–1 victory. His pass to Marco Ureña also set up Costa Rica’s third goal in the game. During the round of 16, Campbell was the fourth of five Costa Rican players to successfully convert his kick in a 5–3 penalty shootout victory over Greece.

On 24 January 2015, Villlarreal announced the signing of Campbell on loan until the end of the season. On 10 May, he scored his first goal for the team, the only goal in a victory over fellow Valencians Elche, which secured sixth place for The Yellow Submarine in La Liga and entry into the following season’s UEFA Europa League.

After returning from his loan at Villarreal, Campbell started his first game of the season in a 2–1 win against Tottenham Hotspur in the League Cup third round on 23 September 2015. On 31 October, due to injuries to Theo Walcott and Alex Oxlade-Chamberlain, Campbell made his first Premier League start for the club against Swansea City at the Liberty Stadium, scoring his first competitive goal in a 3–0 win. Having scored his first Premier League goal, Campbell scored his first Emirates Stadium goal against Sunderland on 6 December, the first for Arsenal in a 3–1 win. The sides faced at the Emirates again in the FA Cup third round on 9 January 2016, with Campbell again scoring Arsenal’s first as they matched the scoreline they achieved just over a month earlier.

On 28 July, in the 2016 MLS All-Star Game in San Jose, California, Campbell won an early penalty when fouled by Laurent Ciman, and took it himself to open a 2–1 victory. On 21 August 2016, Portuguese Primeira Liga side Sporting C.P. signed Campbell on loan until the end of the season.

Campbell joined Real Betis on loan for a second time on 31 August 2017. He went on to pick up an injury in a game against Espanyol in late October 2017. This put him out of action for another three months. Campbell sought to make a comeback to the club’s training squad in late January 2018.

On 17 August 2018, Campbell signed a three-year contract with Frosinone. In January 2019, he was loaned out to León for 18 month after failing to score in 17 games.

In May 2018 he was named in Costa Rica’s 23-man squad for the 2018 FIFA World Cup in Russia.