The potential energy curves (PECs) of X4∑- and B4∑- states of the AlC molecule have been studied using highly accurate internally contracted multireference configuration interaction approach with the Davidson modification. The Dunning's correlation-consistent basis sets, aug-cc-pVnZ (n=D,T,Q,5,6) are used for the present study. To improve the quality of PECs, core-valence correlation and scalar relativistic corrections are considered. Core-valence correlation corrections are calculated with an aug-cc- pCVTZ basis set. Scalar relativistic correction calcualtions are made using the third-order Douglas-Kroll Hamiltonian approximation at the level of a cc-pV5Z basis set. Obvious effect on the PECs by the core-valence correlation and relativistic corrections has been observed. All the PECs are extrapolated to the complete basis set limit. The convergence observations of present calculations are made and the convergent behavior is discussed with respect to the basis set. Using these PECs, the spectroscopic parameters (TeReωeωexeωeyeBe and αe) of the X4∑- and B4∑- states are determined and compared with those reported in the literature. The vibration manifolds are evaluated for each state of non-rotation AlC molecule by numerically solving the radial Schrödinger equation of nuclear motion. For each vibrational state, the vibrational level and inertial rotation constants are obtained, which are in excellent accordance with the experimental findings.