The Apple app store is accessible on iTunes by logging into an iTunes account, then selecting the App Store from a list of purchase options that also include music, movies and TV shows.
As of June 2015, the current version of iTunes is version 12.1.2. In this version, after logging in, an option on the top center bar, below the Apple logo, is iTunes Store. Once the iTunes Store button is clicked, there will be icons in the top left of iTunes representing (from left to right) music, movies, and TV. Clicking on the ellipses to the right of these icons will open a list of further options, with Apps being the fourth option down. Clicking this will open the app store.