How do you work out the private key in a RSA cryptosystem knowing a certain public key? The public key: 

n = 85 and e = 19

I know that n is the product of two prime numbers, now is it right of me to figure them out by process of elimination which I got p or q equals 17 and 5. Or is there a proper method of working this out?
In your case you can factor $N (= 85)$ and use that to compute $\phi(N)$, which in turn allows you to compute $d = e^{-1} = 19^{-1} \text{ mod } \phi(N)$. Factoring $N$ yields $5 * 17 = 85$ which in turn means $\phi(N) = (5-1)(17-1) = 64$. Finally, $d = 27 = 19^{-1} \text{ mod } 64 $.

Note that for most actual instances of RSA $N$ is usually at least 1024 bits, and as such cannot be efficiently factored by pretty much anyone. Hence recovering $d$ from just $(e, N)$ is considered unfeasible. This is critical to the security of RSA. It intuitively makes sense too, if you can recover the private key using only the public key then the private key isn't very private.