Question: I'm trying to understand how you calculate the force that is exerted on your body when running, as well as which forces have the biggest impact.

My understanding is that, given that your mass is constant, the velocity (i.e. 3.5m/s) is the biggest factor in determining the force impacted on the body, however my friend is saying that gravity is the most important factor to consider.

Answer: 
You need to differentiate between the force of gravity or your weight, $mg$ and impact force, which is the result of work required to change the kinetic energy (velocity) of an object. 

When you are running you incur an impact force based on your stopping distance and the kinetic energy you had just prior to impacting the ground with your foot. From the work-energy theorem the net work done on an object equals its change in kinetic energy, or

$$W\_{net}=F\_{ave}d=\frac{mv^2}{2}$$

Where $F\_{ave}$ is your average impact force, $d$ is the stopping distance of your center of mass from the instant of impact to when your downward velocity is zero, which can be a function of how much you bend your knee during the impact with the ground, and any penetration distance into the ground, and $v$ is your velocity just prior to impact. I should point out that this equation neglects the contribution of the positive work done by the force of gravity acting through the stopping distance, since it is typically small compared to the negative work of the average impact force.

You may be interested that the Saucony shoe company reports a runner weighing 670 N (150 lb) can experience impact forces of up to seven times their body weight, or 4700 N (1000 lb) when running.

Hope this helps.
You are 100% right, gravity is a constant, so **of course** velocity is the most important factor, you have to be a very simple person to think otherwise.