Your first step on the road to expanding your Blade Runner horizons should be the 1997 PC point-and-click adventure game developed by Westwood Studios and published by Virgin Interactive. Rather than strictly adapting the events of the film, it crafts an original story that actually runs parallel to the goings-on of Ridley Scott's classic. Players take up the role of detective Ray McCoy, who's just been tasked with hunting down a large group of dangerous Replicants.
This 3D-based adventure does feature film protagonist Rick Deckard, but he isn't given prominent lines or very much of a role in the video game 'sidequel,' as it was billed. It takes place shortly after the film begins and then weaves around and intersects with certain parts of the movie, providing extra context to certain scenes and further expanding the Blade Runner film universe.
The game asks players to question others, react to results of Voight-Kampff tests, and gather information that's ultimately stored in McCoy's Knowledge Integration Assistant (KIA) to automatically organize clues and other information integral to progressing through the game. It's a challenging and satisfying extension of the film, and one that you can pick up right now fairly cheaply if you have DOSBox knowledge and a little patience. Or you can check out one of several hundreds of Let's Play videos instead.