Wilkinson was born in Iowa and moved to Wyoming with his parents in 1945. Wilkinson played high school football in the small town of Greybull, Wyoming. He was scouted as both a football and baseball player out of high school.
Wilkinson attended the University of Wyoming on a football scholarship. After graduation, he was signed as a quarterback by the Toronto Rifles of the Continental Football League. He played 16 games for the Rifles in 1966 and 1967, throwing the ball 287 times for 141 completions with 1,952 yards, 11 interceptions and 18 touchdowns. It was during 1967 he moved to the Toronto Argonauts of the Canadian Football League as a backup quarterback. He was the team's leading quarterback in 1969 throwing for 2,041 yards. In 1970, Wilkinson made way for Don Jonas to lead the Argos, but still got enough playing time to throw for 1,272 yards. Traded to the BC Lions in 1971, he played there for a year before being released prior to the 1972 CFL season. He was picked up by the Edmonton Eskimos and played well enough in the remaining pre-season games to make the team as a backup, but by season's end he was their top passer with 2,475 yards in which he completed 177 out of 268. Wilkinson would play the balance of his career there in a two-quarterback system. First with Bruce Lemmerman and then with Warren Moon. After his first professional seasons, when he was considered to be injury prone, Wilkinson did not miss a game during his 10 years with the Eskimos.
He first led the Eskimos to a Grey Cup victory in 1975, after losses in the championship game in 1973 and 1974. After quarterbacking the Eskimos during one more Grey Cup loss in 1977, Wilkinson became part of the 5-time Grey Cup champion Eskimo dynasty from 1978 through 1981, the last two years acting as a backup to Moon.
Including his time with the Rifles, Wilkinson threw for 24,531 passing yards on 1754 completions from 2,949 attempts and with 137 interceptions to 172 touchdowns. Although not known for his running abilities he also added 1,250 rushing yards on 352 runs with 13 touchdowns.
He retired after the Eskimos' 1981 Grey Cup Championship year. Wilkinson was the first player honoured on the Edmonton Eskimo's "Wall of Fame" at Commonwealth Stadium in 1982.
In 2012 in honour of the 100th Grey Cup, Canada Post used his image on a series of commemorative postage stamps. The image was also used on presentation posters and other materials to promote the Grey Cup game and other celebrations associated with the centennial.