$\newcommand{\Ket}[1]{\left#1\right>}$
I would like to know if there is a systematic way of finding a set of Kraus operators $E_k$ for a quantum channel $\varepsilon$ defined by its action on a density matrix $\rho$ using these properties:
$\varepsilon(\rho) = \sum_{k}E_k\rho E_k ^{\dagger}$
$\sum_{k}E_k^{\dagger} E_k = \mathbb{1}$
I feel like you can "educated" guess the answer but I would like to know if there is a more formal method.
To be more specific, there are 2 different input possibilities for the problem: the first is when the channel $\varepsilon$ is defined by its action on a density matrix $\rho$, and the second is when it is defined by its action on kets.
Let me give an example for both these cases:

Action on $\rho$: find the Kraus operators for the dephasing channel:
$\rho \rightarrow \rho ' = (1p)\rho + p\, diag(\rho_{00},\rho_{01})$ 
Action on kets: find the Kraus operators for the amplitude damping channel, defined by the action:
$\Ket{00} \rightarrow \Ket{00}$, $\Ket{10} \rightarrow \sqrt{1p}\Ket{10} + \sqrt{p}\Ket{01}$
I cannot figure out a method for any of these types of cases even though I know a possibility of Kraus operators for both of these cases. For the dephasing channel:
$E_0 = \sqrt{1p/2}\mathbb{1}$ and $E_1 = \sqrt{p/2}\sigma_z$
and for the amplitude damping channel:
$E_0 = \begin{pmatrix}
1 & 0 \\
0 & \sqrt{1p}
\end{pmatrix} \quad E_1=
\begin{pmatrix}
0 & \sqrt{p} \\
0 & 0
\end{pmatrix}$
Best Answer
you can use the choi isomorphisme:
you apply the Choi map to your channel to obtain the corresponding Choi matrix and then you compute the spectral decomposition of this matrix. The Kraus operators will be the eigenvectors rearranged (vectorization) into a matrix and the weight of each Kraus operator will be the corresponding eigenvalue.