With calculations involving GR & gravity, when are Newtonian mechanics & Newtonian gravity sufficient and when are they not? I understand that Newtonian mechanics is a good approximation of GR but at what extremities are the differences so great that GR must be used. 
I assume it to not be suitable at velocities nearing the speed of light or for black holes and neutron stars. 
 A: People usually talk about things like this in terms of 'characteristic scales' --- which just mean the typical lengths, masses, times, etc at which different phenomena occur or become important.  For special-relativity, the characteristic quantity is the speed of light '$c$'.  When velocities are much slower than the speed of light, i.e. $v \ll c$ (or equivalently $v/c \ll 1$), then special relativity is a small effect.  For gravity, an easy characteristic quantity for the 'strong-field regime' is the Schwarzschild Radius,
$$r_s = \frac{2GM}{c^2} \approx 3 \, \mathrm{km} \left(\frac{M}{M_\odot} \right).$$
If objects have sizes (or are acting at distances) such that $r \gg r_s$ (i.e. $r_s/r \ll 1$) then General Relativity (and general relativistic corrections) is unimportant.   But as @acuriousmind points out, what is or is not "important" or "negligible" depends entirely on the situation.  For gravitational wave detection, for example, $10^{-21}$ is certainly important and not negligible.
A: PPN formalism is the best way unless you are close to very strong gravity like very near the horizon of a sun like (or even 100 times more massive) black hole. It's been used for the in-spiral part of two black holes coalescing. Try finding some on it at https://en.m.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism. You can also use the linearized general relativity equations if you want to figure out gravitational radiation at https://en.m.wikipedia.org/wiki/Linearized_gravity
Finally, which approximation you use depends on the application. For very strong fields near black holes you need to do a numerical simulation. I doubt that'll be the case for you. There may be other approximations for what you need, a simple google search is 'approximations in general relativity'. Since the full theory is highly nonlinear where you start for the base from which you linearized makes a difference. You may also try approximations off the charged Schwarzchild metric, which includes spherical symmetric spacetime (used for orbits around sun and black holes), and there is a solution with charge. See for spherically symmetric space times the orbits (geodesics) at https://en.m.wikipedia.org/wiki/Schwarzschild_geodesics. You will probably need to find the equivalent paper for the charged solution. 
A: Here a more qualitative than quantitative answer:
You don't need a black hole or a neutron star to observe differences between Newton's law of gravity and the real laws of nature. As an example, before General Relativity was discovered, there was an observation, that the orbit of Mercury (more specifically its perihelion) did not behave as expected. Hence already an object with the mass of the sun causes a gravitational field large enough for Newton's laws to break down.
