Suppose I'm standing a few light minutes away from a light emitting object, as soon as the light is turned on, I project myself towards the object with some speed. Will the relative velocity of my speed and $c$ add up to give a velocity higher than speed of light?
No. This will never happen according to Special Relativity and its addition of velocities principle. You might be thinking of Galilean Transformation in which two velocities will add with no limitation. But, these transformations are only special cases of Lorentz Transformation where the following postulate holds true and can not be violated:
As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body. OR: The speed of light in free space has the same value c in all inertial frames of reference.
Take a look at the following source in which you can plug in values for v and u' and yet resultant velocity u will never exceed c: Einstein Velocity Addition So, Nature is bound by a stringent constraint for the velocity of any object.