How related is the distribution of primes to the Riemann Hypothesis?

would a 'formula' or other system that enables you to calculate the distribution

of prime numbers enable mathematicians to solve the Riemann Hypothesis?

There is an exact formula, known as "the explicit formula" of Riemann, for the prime number counting function $\pi(n)$ in terms of the zeros of $\zeta(s)$. (Really it uses a minor modification of $\pi(n)$, extended to positive real values of $n$, but the idea is the same.)

The explicit formula displays an equivalence between asymptotics of the prime number distribution and location of zeros of $\zeta(s)$. Knowledge of the real part of the location of the zeta zeros translates into knowledge of the distribution of primes. The closer the zeros are to the line with real part $1/2$, the better the control over the distribution of primes.

This is all in Riemann's paper approximately 150 years ago, that introduced the Riemann hypothesis. The prime number theorem is equivalent to a demonstration that no zeros have real part equal to $1$, which was done at the end of the 19th century. The infinitude of primes is equivalent to the pole of $\zeta(s)$ at $s=1$, as was shown by Euler.

The difficulty in finding all the zeros is not the lack of a formula, but that the explicit formula relates two complicated sets without proving anything about either set individually. To restrict the location of zeta zeros through a formula for prime numbers, the prime formula would have to be strong enough to estimate $\pi(n)$ with an error of order $n^{1-\epsilon}$ for a positive $\epsilon$, which would be considered an incredible breakthrough. Using Riemann's explicit formula it would be possible to take any argument about the prime distribution and translate it relatively easily into an argument about the zeta function, so it's not the case that formulations in terms of primes are likely to be any more amenable to proof than talking about the zeta zeros. In fact it is usually easier to start from the zeta function.

As lhf writes, there is a strong link between the error estimate in the prime number theorem and the Riemann hypothesis. Indeed, RH is equivalent to a certain bound on this error estimate.

More precisely, the prime number theorem states that $\pi(x)$ (the number of primes $\leq x$) is asymptotic to $\mathop{\mathrm {Li}}(x)$, and the Riemann hypothesis is equivalent to the statement that the error in this approximation is bounded (for large $x$) by $x^{1/2 + \epsilon}$ for any $\epsilon > 0$.

The asymptotic prime number distribution has been known for over a century now. The Riemann Hypothesis is about the error term in that asymptotic equation. In this sense, they are very closely linked.