We predict high-gain x-ray lasing in molecular nitrogen by ultrafast core ionization with an x-ray free-electron laser source. To estimate the spectral and temporal output of this molecular x-ray laser, we solve generalized Maxwell-Bloch equations, keeping track of the electronic and nuclear degrees of freedom. The spectrum of the amplified x-ray emission shows a strong dependence on the gain-length product. Whereas the emission at small gain length is similar to the relatively broad x-ray fluorescence band, the spectrum is determined by a single frequency in the linear gain region. The vibrational wave packet dynamics during the x-ray emission process is examined. By preparation of the initial vibrational quantum state, the x-ray emission frequency can be tuned within the fluorescence band. The present scheme is applicable to other homo- and heteronuclear diatomic systems, thereby extending the spectral range of coherent x-ray radiation sources based on amplification on bound transitions.