We study a solar spicule observed off-limb using high-resolution imaging spectroscopy in the Ca II 8542 Å line obtained with the CRisp Imaging SpectroPolarimeter (CRISP) on the Swedish 1 m Solar Telescope. Using a new version of the non-LTE code NICOLE specifically developed for this problem we invert the spicule single- and double-peak line profiles. This new version considers off-limb geometry and computes atomic populations by solving the 1D radiative transfer assuming a vertical stratification. The inversion proceeds by fitting the observed spectral profiles at 14 different heights with synthetic profiles computed in the model by solving the radiative transfer problem along its length. Motivated by the appearance of double-peak Ca II 8542 Å spicule profiles, which exhibit two distinct emission features well separated in wavelength, we adopt a double-component scenario. We start from the ansatz that the spicule parameters are practically constant along the spicule axis for each component, except for a density drop. Our results support this ansatz by attaining very good fits to the entire set of 14 × 4 profiles (14 heights and 4 times). We show that the double-component model with uniform temperature of 9560 K, exponential decrease of density with a height scale of 1000-2000 km, and the counter-oriented line-of-sight velocities of components reproduce the double-peak line profiles at all spicule segments well. Analyses of the numerical response function reveals the necessity of the inversions of spectra at multiple height positions to obtain height-dependent, degeneracy-free reliable models with a limited number of free parameters.