Finding Kraus operators
The method of determining the Kraus operators of a quantum channel $\Phi: L(\mathcal{X})\to L(\mathcal{Y})$ from the knowledge of its action on a set of inputs is called quantum process tomography. See section 8.4.2 on page 389 in Nielsen & Chuang for details. In particular, see equation $(8.168)$ on page 392 for how Kraus operators are computed. In general, the method requires the knowledge of the action of $\Phi$ on a basis of the input space $L(\mathcal{X})$. For one qubit, this means knowing the action of $\Phi$ on at least four input states.
Recovering $\Phi$ from knowledge of its action on single input
As @gIS points out in the comments, it is in general impossible to recover $\Phi$ from the knowledge of its action on a single input density matrix $\rho\in D(\mathcal{X})\subset L(\mathcal{X})$. However, as you anticipate in your responses, it is possible to place certain restrictions on $\Phi$ that do make such determination possible.
Below, I describe two such sets of restrictions. The first one is motivated by basic linear algebra and is not particularly surprising. The second one comes from quantum information science and is perhaps more unexpected.
Action of $\Phi$ on a basis
Let $d=\dim\mathcal{X}$ and let $X_1, X_2, \dots, X_{d^2}\in L(\mathcal{X})$ denote a basis of the input space $L(\mathcal{X})$. Suppose that in addition to knowing the action of $\Phi$ on $\rho$, we also know the action of $\Phi$ on $X_1, X_2, \dots, X_{d^2-1}$ where $\rho$ is linearly independent from $X_1, X_2, \dots, X_{d^2-1}$. In this case, $\rho, X_1, X_2, \dots, X_{d^2-1}$ is itself a basis of $L(\mathcal{X})$ and so we can recover $\Phi$.
A special case of this occurs when $d=1$ and $\Phi$ is a state preparation channel which is indeed fully described by its sole output state.
Action of $\Phi$ on a full rank bipartite input
It turns out that with the help of an auxiliary system, a quantum channel can be recovered from its action on a single input. More precisely, if we know the action of $\Phi\otimes I$ on a joint state $\rho\in D(\mathcal{X}\otimes\mathcal{X})$ whose Schmidt rank is $d^2$ then we can recover $\Phi$ from the knowledge of $(\Phi\otimes I)(\rho)$ alone.
The method is called entanglement-assisted process tomography (EAPT) or ancilla-assisted process tomography (AAPT). In the former, $\rho$ is entangled, in the latter it is not. The latter is possible because, perhaps somewhat surprisingly, there do exist unentangled states of full Schmidt rank. See description of figure 1 c in this paper for an example.
