In order to find eigenvalues, you will become well acquainted with:
Ax = Lx (where L is lambda);
Similarly, (A - LI)x = 0, where I is the identity matrix, and x is the corresponding eigenvector.
The characteristic polynomial is simply the part before the equality (= 0). The characteristic equation is then the whole thing. What you're interested in finding is the determinant of (A - LI)x. Do you have a specific problem that you're stuck on?