Stellar signals are the main limitation for precise radial-velocity (RV) measurements. These signals arise from the photosphere of the stars. The m/s perturbation created by these signals prevents the detection and mass characterization of small-mass planetary candidates such as Earth-twins. Several methods have been proposed to mitigate stellar signals in RV measurements. However, without precisely knowing the stellar and planetary signals in real observations, it is extremely difficult to test the efficiency of these methods. The goal of the RV fitting challenge is to generate simulated RV data including stellar and planetary signals and to perform a blind test within the community to test the efficiency of the different methods proposed to recover planetary signals despite stellar signals. In this first paper, we describe the simulation used to model the measurements of the RV fitting challenge. Each simulated planetary system includes the signals from instrumental noise, stellar oscillations, granulation, supergranulation, stellar activity, and observed and simulated planetary systems. In addition to RV variations, this simulation also models the effects of instrumental noise and stellar signals on activity observables obtained by HARPS-type high-resolution spectrographs, that is, the calcium activity index log(R'hk) and the bisector span and full width at half maximum of the cross-correlation function. We publish the 15 systems used for the RV fitting challenge including the details about the planetary systems that were injected into each of them (data available at CDS and here: https://rv-challenge.wikispaces.com.