Condition number reflects the sensitivity of "something" with respect to the change of data, in this case the perturbations of the matrix $A$. While the condition number $\|A\|\|A^{-1}\|$ characterizes, e.g., the sensitivity of the inverse and solutions of linear systems with $A$, the sensitivity of eigenvalues depends on different factors.
In the simpler case with distinct eigenvalues (where the eigenspaces are one-dimensional), the condition number of a given eigenvalue $\lambda$ with the right and left eigenvectors $x$ and $y$, respectively, is given by
$$
\kappa(\lambda)=\frac{\|x\|_2\|y\|_2}{|x^*y|}.
$$
That is, if the left and right eigenvectors coincide (as it is, e.g., for normal matrices), the eigenvalues are well conditioned ($\kappa(\lambda)=1$).
However, if the eigenvectors are nearly orthogonal, the eigenvalues are very sensitive.
You can easily construct an ill-conditioned (w.r.t. to $\mathrm{cond}(A)$) normal matrix (think about a diagonal matrix with large magnitude differences of the diagonal entries). You can also construct a well-conditioned matrix with sensitive eigenvalues, e.g., the Grcar's matrix (it can be generated by the gallery command in MATLAB). It is fairly well-conditioned w.r.t. to $\mathrm{cond}(A)$ but has extremely sensitive eigenvalues. E.g., for $n=100$, $\mathrm{cond}(A)\approx 3.59$ while the "least sensitive" eigenvalue has the condition number of about $10^6$.
A useful source of results could be, e.g., this book.
