He attended the University of Texas at Austin, where he played both football (under coaching legend Darrell Royal) and baseball, leading the Southwest Conference in batting average.

[5] Alvis was called up to Cleveland at the end of the 1962 season, and he played his first major league game on September 11, 1962.

He enjoyed single-season career-high numbers in batting average (.274), RBI (67), runs (81), hits (165), doubles (32) and triples (7).

All the players and crew who flew with him from Minnesota before his diagnosis and hospitalization received cautionary treatment to prevent the disease from spreading.

[10] His condition improved with treatment within a day, but he was placed on the disabled list and Chico Salmon was called up to replace him.

[8] Alvis turned in a solid 18 home run performance in 1966 and led the team with 21 in 1967,[16] batting .245 and .256 respectively in those years.

His batting average fell to .223 as a full-time player in 1968, and he was relegated to spot duty with Cleveland in 1969, appearing in only 66 games.

[8] Alvis was traded along with Russ Snyder from the Indians to the Brewers for Roy Foster, Frank Coggins and cash during spring training on April 4, 1970.

His Cleveland teammate Sam McDowell observed that Alvis was the most decent person he met in baseball, and the most disciplined and well-conditioned player on the team; rarely drinking alcohol (unusual for the time).

[2] He and his wife Frances Mae (Eddy) Alvis were married on August 23, 1958, and had two sons, Max, Jr. and David.