A native of Loganville, Georgia, Nkemdiche attended Grayson High School, where he was a three-sport athlete in football, basketball, and track. He played as a defensive end and running back for the Grayson Rams high school football team. In his junior season, he registered 59 tackles and 18 sacks on defense while also running for 528 yards and scoring 17 touchdowns on offense. Grayson High School went undefeated throughout the season, finishing 15–0 with a GHSA 6A state title win over Marietta Walton. In his senior year, Nkemdiche registered 59 tackles, 12 tackles for loss and seven sacks on defense, while on offense, he had 235 total rushing yards and 10 touchdowns. Early in the season, Grayson defeated Dalvin Cook's Miami (FL) Central 35–3 in a game that was nationally televised on ESPN. The Rams went on to a 10–2 season record, and were upset by Mitch Hyatt's North Gwinnett in the second round of the GHSA 6A state playoffs. Nkemdiche was the Gwinnett Daily Post Defensive Player of the Year. He finished his career at Grayson with 41 career sacks. He was named first-team All-American as a junior by MaxPreps, as well as second-team as a senior.
In addition, Nkemdiche in track & field during his junior season. At the 2011 Gwinnett County T&F Championship, he earned second-place finishes in both the shot put, with a throw of 13.41 meters (43 ft, 8 in), and the discus, with a throw of 37.98 meters (124 ft, 5 in). In addition, he had marks of a 350-pound max bench press, 335-pound power clean, 500-pound squat, and a 4.56-second 40-yard dash time.
Regarded as a five-star recruit by both Rivals.com and Scout.com, Nkemdiche was not only considered the No. 1 defensive line prospect of his class, but also the consensus No. 1 player overall. He was only the fourth player since 2002 to be the consensus No. 1 prospect, along with Jadeveon Clowney, Vince Young, and Ernie Sims. Nkemdiche went wire-to-wire as the No. 1 overall player in the Rivals.com Top100, also the first since Clowney.
Robert and his older brother, Denzel, originally planned attending college together. After Denzel Nkemdiche signed with Ole Miss, Robert was believed to follow. However, he instead committed to Clemson on June 14, 2012. In October 2012, ESPN The Magazine reported that Nkemdiche was not firm on his non-binding verbal commitment, and was still considering other schools, including Ole Miss. Consequently, Nkemdiche decommitted from Clemson in November 2012, re-considering a number of schools including Clemson, Ole Miss, Alabama, and Georgia. By January 2013, he narrowed his list down to just Ole Miss and Louisiana State.
Giving her opinion on their sons' college selection, Nkemdiche's mother said she would "love for them to play together, as well as have the power of two." That way, she would not have to worry about going to different games—taking pressure off of her shoulders. Also, on top of the family tie to Ole Miss, Nkemdiche was said to have a close relationship with the head coach of the Ole Miss football team, Hugh Freeze. In November 2012, Denzel Nkemdiche told reporters that Robert personally told him that he was going to commit to Ole Miss. Robert Nkemdiche indeed committed to Ole Miss on National Signing Day to play college football under head coach Hugh Freeze.
One of the most highly touted recruits in the program's history, Nkemdiche arrived at Ole Miss with plenty of hype, being named SEC’s Top Impact Freshman by Sporting News before the season. He played in 11 games with 10 starts (six at defensive end and four at defensive tackle), and amassed 34 tackles (25 solo), 2.0 sacks, two pass breakups, and three quarterback hurries on the 2013 season. Nkemdiche made his college debut against conference opponent Vanderbilt, posting two tackles with a tackle-for-loss and a pass break-up. He missed the Louisiana State and Idaho games with a strained hamstring. After the season, Nkemdiche was named Freshman All-SEC by the league's coaches, and a First team Freshman All-American selection by Athlon and College Football News. As a sophomore, Nkemdiche started every game at the defensive tackle position on an Ole Miss defensive line that led the nation in scoring defense (16.0 ppg) and lead the SEC in tackles for loss (7.6/game). Nkemdiche himself contributed 35 tackles with 4.0 tackles-for-loss, 2.0 sacks, and a pass breakup on the season. His best game was against in-state rival Mississippi State, in which he was credited for a career-high seven tackles, of which three were solo, and one tackle-for-a-loss. Nkemdiche was a semifinalist for the Lombardi Award and was named All-SEC first team by Associated Press (AP), as well as All-America second team by AP, Sporting News, USA Today, and CBS Sports.
After an incident involving marijuana (see below) in December 2015, Nkemdiche was suspended for the 2016 Sugar Bowl against Oklahoma State. He announced his intention to enter the 2016 NFL Draft.
On April 28, 2016, the Arizona Cardinals selected Nkemdiche in the first round with the 29th overall pick in the 2016 NFL Draft. He only recorded one tackle during his rookie season.
Nkemdiche was born in Atlanta, Georgia to Nigerian immigrants Beverly and Sunday Nkemdiche. His father, a cardiologist, has reportedly applied for a researcher position at University of Mississippi Medical Center in Jackson, Mississippi. Nkemdiche's mother, a politician, returned to Nigeria in 2009 and as of September 2011 represents Onitsha South in the Anambra State House of Assembly. His older brother, Denzel Nkemdiche, plays linebacker for Ole Miss.
On December 12, 2015, Nkemdiche fell from a fourth-floor window of the Grand Hyatt Atlanta in Buckhead. “It appears that the victim (Nkemdiche) broke the window and then walked approximately 15 feet and climbed over a wall and fell approximately 15 feet,” read the Atlanta Police Department report. Nkemdiche was conscious and in stable condition when officers reported to the scene, and was immediately transported to Grady Memorial Hospital. Police later found seven marijuana cigarettes in Nkemdiche's room, leading him to be charged with drug possession.