I must find the eigenvalues of the following system: $\hat H = V (|\alpha \rangle \langle \beta| +|\beta\rangle\langle\alpha|)$, where $\langle \alpha|\alpha \rangle=\langle\beta|\beta\rangle=1 $, $\langle \alpha|\beta\rangle=0, $ $\hat H$ is Hermitian and $V\in \mathbb{R}$.
We were given the hint of seeking the eigenvectors in the form $C_1|\alpha\rangle + C_2 |\beta\rangle$, so I did the following work.
$\hat H |n\rangle = E|n\rangle,$ and, say, $|n\rangle =C_1|\alpha\rangle + C_2 |\beta\rangle.$
\begin{align*}\therefore \hat H |n\rangle&= V\Big(|\alpha \rangle \langle \beta| +|\beta\rangle\langle\alpha|\Big)\Big(C_1|\alpha\rangle + C_2 |\beta\rangle\Big) \\ &=V\Big(C_1 |\alpha \rangle \langle \beta|\alpha \rangle +C_2 |\alpha \rangle \langle \beta|\beta \rangle + C_1|\beta\rangle \langle \alpha|\alpha\rangle +C_2|\beta\rangle\langle\alpha|\beta\rangle\Big). \end{align*}
Using the relations given at the start, this
$ = VC_2|\alpha\rangle + VC_1 |\beta\rangle. $
This is as far as I got. Is this the answer to the question? Can I say that the eigenvalues for $\hat H$ are $VC_2$ and $VC_1$?
Good so far, but you haven't invoked the eigenvalue/eigenvector equation, yet!
(The short answer to your question at the end is, "No.")
You must have \begin{align*} VC_2|\alpha\rangle + VC_1 |\beta\rangle&=E\left(C_1|\alpha\rangle + C_2 |\beta\rangle\right). \end{align*} Because of orthogonality, this can only occur if $VC_2=EC_1,$ and $VC_1=EC_2.$ That is, unfortunately, only two equations in three unknowns. I imagine you could say that $\langle n|n\rangle=1,$ and that would produce conditions on the $C_j$'s that would get you to the finish line. Can you continue now?