We present a comprehensive experimental and theoretical investigation of the thermodynamic properties: specific heat, magnetization, and thermal expansion in the vicinity of the field-induced quantum critical point (QCP) around the lower critical field Hc1≈2 T in NiCl2-4SC(NH2)2. A T3/2 behavior in the specific heat and magnetization is observed at very low temperatures at H=Hc1, which is consistent with the universality class of Bose-Einstein condensation of magnons. The temperature dependence of the thermal expansion coefficient at Hc1 shows minor deviations from the expected T1/2 behavior. Our experimental study is complemented by analytical calculations and quantum Monte Carlo simulations, which reproduce nicely the measured quantities. We analyze the thermal and the magnetic Grüneisen parameters, which are ideal quantities to identify QCPs. Both parameters diverge at Hc1 with the expected T−1 power law. By using the Ehrenfest relations at the second-order phase transition, we are able to estimate the pressure dependencies of the characteristic temperature and field scales.