What is the suitable theory to find the potential of $N$ particles?

No, this is not correct. Which theory you use depends on the energy scale you're working on, not on the number of particles. Density functional theory can (in principle) treat hundreds of atoms, but is often used in a non-relativistic context.

In principle, finding the potential is trivial (if you're working with e.g. systems of protons and electrons at reasonably low energy): The potential is just the sum of the Coloumb interactions between the particles.

However, this expression is not very helpful, as you are no closer to solving the Schrödinger equation (and solving it becomes a hopeless endeavor once you have more than a few particles).

So what to do? Well there are many possible approaches, depending on how many particles you have.

Focusing on density functional theory (DFT), it works by reformulating the Schrödinger equation in terms of electron density, and then deriving an expression for the resulting potential. This leads to a term called the exchange-correlation functional, for which sadly no closed-form expression is available.

However, there are a host of techniques for approximating this term. Which of them you use is part of the art of DFT.

Of course, once you start thinking about higher energies, other physical theories become relevant.