The coefficient of lift is a number associated with a particular shape of aerofoil, and is incorporated in the lift equation to predict the lift force generated by a wing using this particular cross section.
Note that the lift equation does not include terms for angle of attack - that is all wrapped up within the description of airfoil geometry. The graph for lift coefficient vs. angle of attack follows the same general shape for all aerofoils, but the particular numbers will vary. The graph shows a linear increase in lift coefficient with increasing angle of attack, up to a maximum point, after which the lift coefficient falls away rapidly. This is known as the stall angle of the aerofoil.
The coefficient of lift is a dimensionless number.
Note that in the graph here (which is generic, not representing any particular aerofoil), there is still a small but positive lift coefficient with angles of attack less than zero. This is true of many actual designs; in particular those designs including asymmetry or camber: a curvature of the aerofoil. As a consequence of the inertia of the air, the trailing edge of an airfoil will deflect air downwards more than the leading edge can deflect air upwards. The trailing edge of cambered airfoils has a downward tilt even when the airfoil as a whole is tilted to zero angle of attack. As a result, a cambered airfoil will deflect air downwards even at zero attack angle, and it will create a small amount of lift at zero and at small negative angles.