You misunderstand special relativity.
First, you would not be able to travel at the speed of light relative to some inertial frame, as this would require an infinite amount of energy. So let's instead say we are moving at a constant velocity near the speed of light relative to the Earth, and that as we go by someone on earth shines a light. What do we see?
For special relativity, it is first assumed that the speed of light will be measured the same for any inertial reference frame. We start here. It is not a consequence of SR, it is actually an assumption. So then you seem to ask from a classical interpretation "Well if I am going fast enough I will be moving closely along with the light that was just emitted from Earth. Then let my consider time dilation." This is not the case.
You are applying a classical reasoning and then using SR. What would actually be happening is that the effects of both time dilation and length contraction would apply to whatever you are using to measure the passage of the light by you, and these effects would "cancel out" so that you still measure the same speed of light. But this is just showing the internal consistency of the theory. Time dilation and length contraction are a consequence of the fact that there is no "absolute frame of reference". All inertial reference frames should observe the same physics.
Einstein's time dilation theory says that the faster you travel through space,the slower you move through time(that means that our physical and biological processes slow down)
I want to comment on this as well, since there are things that need to be cleared up. In SR we are dealing with inertial frames. In my own inertial frame, I observe myself to be at rest. My biology ticks away at "normal" time.
Furthermore, there is no defined "faster travel through space", since this assumes an absolute reference frame. What you can say is that if someone else were to see you moving "faster through space" relative to them, then they would see your clock ticking slower. They would see your biology slow down.
You can't just start moving faster relative to something else and expect your own clock to tick slower. That is not what happens.