Cu_xBi₂Se₃ is a superconductor that is a potential candidate for topological superconductors. We report our laser-based angle-resolved photoemission measurement on the electronic structure of the Cu_xBi₂Se₃ superconductor, and a detailed magneto-resistance measurement in both normal and superconducting states. We find that the topological surface state of the pristine Bi₂Se₃ topological insulator remains robust after the Cu-intercalation, while the Dirac cone location moves downward due to electron doping. Detailed measurements on the magnetic field-dependence of the resistance in the superconducting state establishes an irreversibility line and gives a value of the upper critical field at zero temperature of ∼4000 Oe for the Cu_0.3Bi₂Se₃ superconductor with a middle point T_c of 1.9 K. The relation between the upper critical field H_c2 and temperature T is different from the usual scaling relation found in cuprates and in other kinds of superconductors. Small positive magneto-resistance is observed in Cu_0.3Bi₂Se₃ superconductors up to room temperature. These observations provide useful information for further study of this possible candidate for topological superconductors.