This is how I would approach this question if it were in one of my former engineering classes.
I believe Chris Rock meant in a sort of half-assed answer is the following:
There are forces that are a result of friction (this would stay constant) and gravity (this varies with height). Because we are dealing with work, W = Fd, we can also use potential and kinetic energy (for you kiddies check the units they match!).
In the case with running, as someone previously stated, both feet leave the ground resulting in a change in height (thanks for the back up by the way). Taking into consideration the force due to gravity or F=mg and taking the difference in height = h we have PE = mgh. Knowing that forces act independent of each other, in other words forces in the x-direction (back and forth if you will) have no effect of forces in the y-direction (up or down if you will) we can state that the KE required for the y-direction is where KE=PE or in other words 0.5mv^2=mgh. Therefore we can determine the KE require used during stride upward.
KE forward would simply be Total Energy = KE + Frictional Forces*distance or in this case 1mile (Ffriction*d). It should be noted that the energy exerted to move forward should be use to overcome friction which brings your overall x-direction velocity down. If you can determine your velocity without friction perhaps that would be easier, but then again we dont have those types of surfaces in reality.
For simplicity I purposely disregarded geometry, but of course this SHOULD be included for both the x and y-directions.
Knowing the Law of Conservation of Energy, which states that energy can not be created or destroyed, it is safe to assume that the KE required to move at a specific pace would be the same energy the body must produce. In other words the KE from running (external) = Energy in Body (internal).
This is not a perfect explanation, but it is clear that walking velocity is much slower than running velocity and that there is a significant difference considering velocity in squared.
If you have the ability to get a physics book, I would suggest making a few assumptions such as the angle at which you leave the ground, using a point source (the center of your bodies gravity) the difference in height between the point source and above the ground, forces of friction from both shoes and ground, and the velocity at which you wish to run. Some of these you dont have to assume because you can measure them on your own. To have a better understanding drawing a diagram would help determine your forces and trajectory.