Given the PDE $\frac{\partial\rho}{\partial t} + 4 \frac{\partial\rho}{\partial x}= 0 $ determine the characteristics.
I understand how to solve this PDE using the method of characteristics as $\rho=f(x-4t)$, but I do not understand what is meant by "determine the characteristics".
Which equations are the characteristics?
If you're already solving the PDE with the method of characteristics, than you're computing the "characteristics"; the solution is a union of characteristic curves.
They are integral curves of a vector field which the solution is tangent to at every point. In other words, the solution is constant along them and the equation reduces to a ODE along the characteristic curves.
In your case, the curve $x(s), t(s)$ such that $$ \dfrac{dx}{ds} = 4 \\ \dfrac{dt}{ds} = 1 $$