RSA How to select a value K such that $e d = 1 + k \varphi(N)$ holds
You apply the extended Euclidean algorithm to $e$ and $\phi(N)$ (which have to have gcd equal to $1$) and we get $x,y \in \mathbb{Z}$ such that

$$xe + y\phi(N) = 1$$

The $x$ (taken modulo $\phi(N)$, if needed) is the $e$ you are looking for. The $k$ is totally irrelevant for encryption/decryption, but it's actually the $y$ in the above equation (rewrite your equation and see).