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