The multi-configuration time-dependent Hartree-Fock (MCTDHF) approach for the description of correlated few-electron dynamics in the presence of strong laser fields is introduced and a comprehensive description of the method is given. Total ionization and electron spectra for the ground- and first excited ionic channels are calculated for one-dimensional model systems with up to 6 active electrons. Strong correlation effects are found in the shape of photoelectron peaks and the dependence of ionization on molecule size.