An eigenvector of a linear operator is a nonzero vectors s.t. the operator multiplies it by a scalar. It's existence for a complex finite dimensional operator can be proven without the determinant.

The minimal and the characteristic polynomials can be defined and described without the determinant.

Where do you think you need the determinant?

Well at some point you may want to actually calculate the eigenvalues. Unless you just dive in and try and compute the characteristic polynomial directly.