Also you can use the hardcore table like function to find the derivative if you do not use the interpolation function:

(*[[All, 2]]* takes all rows and pick up the second column only/ time step = 1*)
ListConvolve[{1, -1}, data[[All, 2]]] 

assuming data is

data =  {{0, 0.1425`}, {1, 0.1425`}, {2, 0.1425`}, ..., {99, 0.523363636`}}

What you had is this

dqdt = Do[Print[D[q[t], t]], {t, 0, 100}]

Which does not work, since "t" becomes a number, and one can't take derivative with respect to a number. You could write

dqdt = Do[Print@Evaluate[D[q[t], t] /. t -> i], {i, 0, 100}]

But it is easier to write

lst = (D[q[t], t] /. t -> #) & /@ Range[0, 100]

Or just take the derivative outside once, and evaluate it

D[q[t], t];
lst = (% /. t -> #) & /@ Range[0, 100]

had quick look. You already found an interpolation function called q. There nothing more to do. In Mathematica, you can differentiate and integration these interpolation functions like any other function. So all you have to do is

D[q[t], t]

to obtain the derivative. This will give you another interpolation function which is the derivative of q.