Density functional theory (DFT) is a type of electronic structure calculation that has rapidly gained popularity. In this article, we provide a step-by-step demonstration of a DFT calculation by hand on the helium atom using Slater's X-Alpha exchange functional on a single Gaussian-type orbital to represent the atomic wave function. This DFT calculation is compared and contrasted with a Hartree-Fock calculation on the same system. This information is accessible by advanced undergraduate students studying physical chemistry or calculus-based physics.