The numerical renormalization group (NRG) is employed to study a double quantum dot (DQD) system consisting of two equivalent single-level dots, each coupled to its own lead and with a mutual capacitive coupling embodied in an interdot interaction U', in addition to the intradot Coulomb interaction U. We focus on the regime with two electrons on the DQD, and the evolution of the system on increasing U'/U. The spin-Kondo effect arising for U' = 0 (SU(2) x SU(2)) is found to persist robustly with increasing U'/U, before a rapid but continuous crossover to ( a) the SU( 4) point U' = U where charge and spin degrees of freedom are entangled and the Kondo scale is strongly enhanced; and then (b) a charge-Kondo state, in which a charge-pseudospin is quenched on coupling to the leads/conduction channels. A quantum phase transition of Kosterlitz-Thouless type then occurs from this Fermi liquid, strong coupling (SC) phase, to a broken symmetry, non-Fermi liquid charge ordered (CO) phase at a critical U'(c). Our emphasis in this paper is on the structure, stability and flows between the underlying RG fixed points; on the overall phase diagram in the (U, U')-plane and evolution of the characteristic low-energy Kondo scale inherent to the SC phase; and on static physical properties such as spin- and charge-susceptibilities ( staggered and uniform), including universality and scaling behaviour in the strongly correlated regime. Some exact results for associated Wilson ratios are also obtained.