We present a new method for the computation of electronic excited states of molecular systems. This method is based upon a recent theoretical definition of multiconfiguration excited states (due to one of us, see M. Lewin, Solutions of the Multiconfiguration Equations in Quantum Chemistry, Arch. Rat. Mech. Anal. 171 (2004) 83-114). Contrarily to previously used methods, our algorithm always converges to a stationary state of the multiconfiguration model, which can be interpreted as an approximate excited state of the molecule. We also compare our method to other approaches and give some explanation of the unsatisfactory behaviours which are sometimes observed when using the latter. The definition is variational. To compute the first excited state, one has to deform paths on a manifold, like this is usually done in the search for transition states between reactants and products on potential energy surfaces. We propose here a general method for the deformation of paths which could also be useful in other settings. Numerical results for the special case of two-electrons systems are provided. We compute the first singlet excited state potential energy surface of the $H_2$ molecule, and give some numerical results concerning Helium-like atoms.