I have to show that $$R_k\cong\mathbb{Z}[X]/(X^2-k)$$ for $R_k= \left\{\left( {\begin{array}{cc} a & b \\ kb & a \\ \end{array} } \right)\ \middle| \ a,b\in \mathbb{Z} \right\} , k\in\mathbb{Z}$
I know I have to find an homomorphism $\phi: \mathbb{Z}[X] \rightarrow R_k$ such that $ker(\phi) = (X^2-k)$ and that $\phi$ needs to be surjective.
I think I need to do something with the fact that the ideal $(X^2-k)$ is the $0$ matrix in $R_k$. But I don't know how to construct this homomorphism. Does someone have any tips?
You won't get anywhere, because your matrix is wrong. I supsect that you meant $\begin{pmatrix}a & b \cr kb & a\end{pmatrix}$.
You need to find a surjective ring morphism $f:\mathbb{Z}[X]\to R_k$ such that $f(X^2-k)=0$, and apply the first isomorphism theorem.
Since $f $is a ring morphism $f(1)=1$, and then $f(-1)=-1$. You can deduce easily that $f(k)=k I_2$. (You may also use the fact that there is only one ring morphism form $\mathbb{Z}$ to any ring, if you prefer).
Then your matrix $M=f(X)\in R_k$ needs to satisfy $M^2=kI_2$, since $f(X^2-k)=0$
Don't you see a matrix in $R_k$ which satisfies that ?
If you still don't see, you could apply Cayley Hamilton to your $2\times 2$ matrix to have a educated guess...