Johnson received his BS in physics in 1982 from the Wright State University, and his MS in physics in 1984, and his PhD in medical biophysics in 1990, both from the University of Utah, in Salt Lake City.

From 1985 to 1989, Johnson was an assistant professor of physics at Westminster College (Utah).

In 1990, Johnson joined the University of Utah, first as a research assistant professor of internal medicine at the Nora Eccles Harrison Cardiovascular Research and Training Institute (CVRTI) and has held faculty positions in the departments of mathematics, bioengineering, physics, and computer science.

In 1992 Johnson founded the Scientific Computing and Imaging (SCI) research group at the University of Utah, which has since grown to become the Scientific Computing and Imaging Institute (SCI Institute) employing over 200 faculty, staff, and students.

In 2013 Johnson was elected to the board of directors of the Computing Research Association (CRA) and in 2012 Johnson was selected as a member of the executive committee of the IEEE Computer Society Technical Committee on Visualization and Graphics.