In high school, with help from a church youth counselor who took him in and eventually adopted him, he improved his grades, became a basketball star as a sophomore at Community Christian in Wilson, North Carolina,[6] and at Charis Prep in Wilson, North Carolina, where he played as a high school junior and senior and earned a basketball scholarship at Miami.

In 2009, he played tight end and finished the season with 17 receptions for 213 yards and five touchdowns in appearances in 13 games.

[7][12] According to scouts, Graham was "extremely athletic, with an outstanding combination of size and speed for the tight end position," yet he was considered very raw since he only had one year of college football experience.

[17] In his rookie season in 2010, Graham started in only five games, playing behind veteran tight end Jeremy Shockey.

[19] On December 19, against the Baltimore Ravens in Week 15, Graham recorded two receiving touchdowns for his first multi-touchdown game in the 30–24 loss.

[22] Graham led all National Football Conference (NFC) rookie tight ends in major receiving categories.

[32] During the Saints' Week 17 win over the Carolina Panthers, Graham broke Kellen Winslow's NFL record of 1,290 receiving yards in a season by a tight end.

However, the record was broken later that day by New England Patriots tight end Rob Gronkowski, who finished the season with 1,327 yards.

[36] In the Wild Card Round against the Detroit Lions, Graham finished with seven receptions for 55 yards and a touchdown in the 45–28 victory.

[54] During the second divisional matchup against the Atlanta Falcons, Graham became known for his trademark touchdown celebration where he would slam dunk the football over the goalpost, a reference to his basketball background.

[55] Prior to the 2014 season, the NFL banned goalpost dunks and made it punishable as "unsportsmanlike conduct," which results in a penalty and a fine.

[56] Graham became the first player penalized under the new rule and was fined $30,000 for dunking during the preseason game against the Tennessee Titans in August 2014.

[57][58] On December 8, against the Carolina Panthers, Graham recorded his fifth game with two receiving touchdowns on the season in the 31–13 victory.

[64] In the Wild Card Round, a 26–24 victory over the Philadelphia Eagles, Graham had three receptions for 44 receiving yards.

No such competing offer sheet was signed, but negotiations between Graham and the Saints were complicated by disagreement as to whether Graham should be treated for bargaining purposes as a tight end (his official position) or as a wide receiver (where he often lines up during games), since the difference in applicable compensation under the franchise tag is more than $5 million for the year.

On July 15, the last day for a new multiyear contract to be negotiated under league rules, the Saints and Graham agreed on a new four-year deal with $21 million guaranteed and a total value of $40 million over four years, making Graham the highest-paid tight end in the league at the time.

He scored the second late in the fourth to give the Saints a three-point lead, but the team eventually fell in overtime.

[100] Graham's ten receiving touchdowns set a Seahawks franchise record for a tight end in a single season.

[102][103] In the 2018 season, Graham was in a position group with fellow tight ends Lance Kendricks, Marcedes Lewis, and Robert Tonyan.

[112] In the Divisional Round of the playoffs against his former team, the Seahawks, Graham caught three passes for 49 yards, including a nine-yard reception for a first down which sealed a 28–23 Packers win.

[116] In the 2020 season, Graham was the primary tight end and shared targets with rookie Cole Kmet.

[117] Graham made his debut with the Bears in Week 1 against the Lions and caught three passes for 25 yards and a touchdown during the 27–23 win.

[121] In the Bears' only playoff game of the season, a 21–9 loss to the Saints, Graham hauled in a 19-yard one-handed touchdown against the team that drafted him as time expired.

[126] Prior to their second preseason game against the Los Angeles Chargers, Graham was arrested on August 18, 2023, in Newport Beach, California, on suspicion of being under the influence of narcotics and obstructing a police officer.

It was later revealed that Graham was disoriented and experiencing a medical episode, which team doctors suspect was caused by a seizure.

[127] On September 24, 2023, Graham caught his first reception in 21 months on an 8-yard touchdown pass from Derek Carr in an 18–17 loss to the Packers.

[131] In 2020 and 2021, his work with the foundation earned him a nomination from the Chicago Bears as their nominee for the Walter Payton Man of the Year Award.

[132] Graham is a pilot and became co-chairman, along with aerobatic champion Sean D. Tucker, of the Experimental Aircraft Association (EAA) Young Eagles program in July 2018, which introduces children between the ages of 8 and 17 to general aviation.

[133] In May 2023, Graham was hit by a car while bicycling in Miami, suffering a laceration on his calf that required stitches.

"[134][135] On August 19, 2023, Graham was taken into custody in Newport Beach, California reportedly due to a medical condition related to seizures.