CL is a function of angle of attack. It's not a function of speed (again, below the transonic region). You don't see CL plotted vs. speed - it's vs. AOA. Yes, in level flight, lift stays constant - pretty much by definition. But if you speed up, the CL doesn't change, the AOA does - which moves the CL to a different point on the CL/AOA curve.
CL is a dimensionless term that measures the ability of a particular body shape to generate lift. You combine that with density and velocity squared term to determine the amount of lift.