He began his career with the California Angels in 1988 but was a streaky hitter and was traded to the Milwaukee Brewers in 1991. After putting up only average numbers with Milwaukee, he was traded to the new expansion team, the Colorado Rockies. It was with the Rockies where he really shone as a player, using the home-run friendly Mile High Stadium to his advantage.
Bichette hit the franchise's first home run in their second game in 1993, against the New York Mets at Shea Stadium. He finished the Rockies' first season with a 21 home runs and a .310 batting average, his personal best for both at the time. Bichette also hit his first home run at the newly-constructed Coors Field, a fourteenth inning smash against the Mets that secured an opening day victory for the Rockies in 1995. Bichette had his best season in 1995, coming very close to the Triple Crown with a .340 batting average, 40 home runs and 128 RBIs and barely lost the MVP voting to the Cincinnati Reds' Barry Larkin.
Bichette began having knee problems in 1996, but was still successful as a hitter, with a .316 average, 31 home runs and 141 RBIs. And over the next three seasons, he hit 26, 22 and 34 home runs, respectively. But by the end of the 1999 season, his production was beginning to drop and the Rockies dealt Bichette to the Cincinnati Reds. However, his fieldwork was suffering tremendously and Bichette was eventually traded to the Boston Red Sox for a season and a half and then the Los Angeles Dodgers. Bichette retired before ever playing a game with the Dodgers on March 22, 2002.
In August 2004, Bichette rejoined professional baseball as a pitcher and first baseman for the Atlantic League's Nashua Pride. It took little time for Bichette to play back into form as he won the Atlantic League's Player of the Month award for August (his first full month back). He completed the month with a .361 average and 13 homers. His best game was on August 28th when he batted 4-for-5 with two homeruns, eight RBIs and his first professional save.
In August 2005, Bichette's son Dante Bichette, Jr. participated in the Little League World Series with his Maitland, Florida team. The Maitland team featured a second son of a former major-leaguer in Tanner Stanley, son of Mike Stanley.