After his sophomore year, Greene transferred to St. Thomas Aquinas High School in Fort Lauderdale, Florida. As a senior, Greene had 43 receptions for 943 yards (21.9 yards per catch) with 13 touchdowns. In the 2010 Class 5A State title game against Tampa’s Plant High School, he recorded six receptions for 148 yards and two touchdowns, including a title-clinching 74-yard pass in the fourth quarter.

Greene enrolled at Florida State University in the fall of 2011 under head coach Jimbo Fisher. As a freshman in 2011, Greene made 38 receptions for 596 yards and seven touchdowns. He was the MVP of the 2011 Champs Sports Bowl in an 18–14 victory over Notre Dame. As a sophomore in 2012, he had 57 receptions for 741 yards and six touchdowns.

As a junior in 2013, Greene led the Seminoles with 76 receptions for 1,128 yards and nine touchdowns, including nine receptions for 147 yards in the 2014 BCS National Championship Game against Auburn. As a senior in 2014, Greene set the school record for career receptions and receiving yards, passing Ron Sellers in both categories. He also set the record for receptions in a season with 99. He also had 1,365 receiving yards and seven touchdowns in 2014. For his career, Greene had 270 receptions for 3,830 yards and 29 touchdowns.

Greene has two sons, Rashad Jr., born in 2014 and Rylan born in 2018.

Greene was drafted 139th overall in the fifth round of the 2015 NFL Draft by the Jacksonville Jaguars. He was the eighth of 11 Florida State Seminoles to be selected in the draft that year. On September 13, 2015, Greene scored his first professional touchdown on a one-yard reception from Blake Bortles in the season opener against the Carolina Panthers. Later in the season, on December 13, 2015, Greene returned a punt 73 yards for a touchdown against the Indianapolis Colts. Overall, he finished his rookie season with 19 receptions for 93 receiving yards and two receiving touchdowns to go along with his one punt return touchdown.

In 2016, Greene appeared in eight games for the Jaguars and recorded five receptions for 32 yards, and lost one fumble. He also contributed 131 punt return yards before going down with an Achilles injury. He was placed on injured reserve on December 3, 2016.

On September 2, 2017, the Jaguars placed Greene on injured reserve.

On September 15, 2018, Greene was released by the Jaguars, but was re-signed two days later. In Week 11 against Indianapolis, with 1:35 left in the game and the Jaguars trailing by three, Greene fumbled his only catch of the game (and only his third of the season), leading to a loss.

In February 2020, Greene signed with the Saskatchewan Roughriders of the Canadian Football League.