You looking for the Jordan Canonical Form or the Rational Canonical Form?
Either way the first step is the same, find the characteristic polynomial. subtract x from the diagonal and calculate the determinant. It should factor into linear terms (if not, you cannot find the JCF unless you pass to the algebraic closure). The roots to this polynomial are the eigenvalues.
You will want to find the minimal polynomial, it must divide the characteristic polynomial (it will have the same roots, but possibly with less multiplicity). Now you either find the elementry divisors (JCF) or the invariant factors (RCF) respectively depending on which form you are looking for.
Once you find the eigenvalues you find the eigenvectors by solving .