I assume you mean time independent perturbation theory. This book by David McIntyre is a great introduction to quantum mechanics (I used it as a supplement to Griffiths'). I especially like how McIntyre uses spin-1/2 as a working example to define the basic concepts of quantum mechanics, and his chapters on perturbation theory are clear and concise.
Below, I describe how to find the first order and second order corrections to energies and energy eigenstates/eigenkets. When $H$ is degenerate, one has to locate the degenerate subspace of $H$ and work with that directly.
For non-degenerate perturbation theory:
Our original Hamiltonian, $H$, and our perturbed Hamiltonian, $H'$, then we have the eigenvalue equation,
$$ H |n^{(0)} = E^{(0)}_{n} |n^{(0)}> $$
where the superscript denotes the order of the correction, i.e. (0) denotes the unperturbed system.
So we may expand the energies in terms of the energy corrections,
$$ E_{n} = E^{(1)}_{n} + E^{(2)}_{n} + ...$$
and the energy eigenkets,
$$ |n> = |n^{(0)}> + |n^{(1)}> + ...$$
So, the elements of the perturbed Hamiltonian are given by
$$ H'_{n,n} = <n^{(0)}|H'|n^{(0)}> = E^{(1)}_{n} .$$
It can be shown, by taking advantage of the properties of the inner product, that the other energies and the eigenket corrections are
$$ E_{n}^{(2)} = \sum\lim_{m \neq n} \frac{|<m^{(0)} | H' | n^{(0)}>|^{2}}{E_{n}^{(0)} - E_{m}^{(0)}},$$
$$ |n^{(1)}> = \sum\lim_{m \neq n} \frac{<m^{(0)} | H' | n^{(0)}>}{E_{n}^{(0)} - E_{m}^{(0)}} |m^{(0)}>.$$
Sorry the math looks bad, I'm not sure how to use Dirac notation on this website.
For degenerate perturbation theory:
So, of course, the degenerate subspace of $H$ in your question is of smaller dimensionality than $H$. This means that not every eigenstate needs a correction! Since you've located the degenerate subspace, now you need to diagonalize the perturbation Hamiltonian $H'$ in the degenerate subspace to obtain the corrections to the original eigenkets. Page 339 of McIntyre's book goes over several examples.