Wave function of a perturbed system
The method is only outlined below, the full details should be in your textbook or lecture notes.  

Your $H^{(1)}$ is just the (small) perturbing potential energy added to the original hamiltonian so you could write $H^{(0)}+\lambda V$.  ($\lambda$ is dimensionless and varies from 0, no perturbation, to 1, full perturbation ).

The Schroedinger equation is now $(H^{(0)}+\lambda V)\varphi_n=E_n\varphi_n$  where $\varphi_n,\; E_n$ are the new wavefunctions and energy. These are expanded as a Taylor series in $\lambda$, e.g.  $\displaystyle \varphi_n \approx \varphi_n^0 + \lambda \frac{d\varphi_n}{d\lambda}+\frac{\lambda^2}{2!} \frac{d\varphi_n^2}{d\lambda^2}+ ..\equiv \varphi_n^{(0)}+\lambda\varphi^{(1)}_n +\lambda^2\varphi^{(2)}_n+..$ and similarly for $E_n$.  

Next this result and that for  $E_n$ are substituted back into the Schroedinger eqn and powers of $\lambda$ factored out. These remaining equations are solved but none now contain $\lambda$; for example the first order term is $\lambda\left((V-E^{(1)}_n)\varphi^{(0)}_n+(H^{(0)}-E_n^{(0)})\varphi^{(1)}\right)$ where in this new notation  $\varphi^{(0)}$ is the original unperturbed wavefunction and similarly for the energy. 

The equation to solve is now $(V-E^{(1)}_n)\varphi^{(0)}_n+(H^{(0)}-E_n^{(0)})\varphi^{(1)}=0$.

After (formally) integrating the equation some fiddling around you find that $\varphi^{(1)}$ is needed. The important step now is to expand this wavefunction in terms of the original basis set of unperturbed wavefunctions $\phi$. This can be done because these form a complete orthonormal set, thus $\varphi^{(1)}= \sum_k a_k\phi_k$ where $a_k$ are the expansion coefficients and these have to be found.  Again after some more fiddling with equations and using the orthonormality of the $\phi_n$ the perturbation energy $E^{(1)}_n = \int \varphi^{(0)*}_nV\varphi^{(0)}_nd\tau = \int\phi^{0*}V\phi^0 d\tau$. 

Notice how only the unperturbed wavefunctions contribute to the energy.  The new wavefunction is $$ \varphi^{(1)}_n=\sum_{n\ne k} \frac{E_n^{(1)}}{E_n^{(0)}-E_k^{(0)}}\varphi^{(0)}$$

which is a linear combination of the unperturbed wavefunctions with the $a$ coefficients as ratio of energies.