We address symmetry-breaking bifurcations (SBBs) in the ground state (GS) and dipole-mode (DM) solitons of the one-dimensional linearly coupled NLS equations, modelling the propagation of light in a dual-core planar waveguide with the Kerr nonlinearity and two types of PT-symmetric potentials. The PT-symmetric potential is employed to obtain different types of solutions. A supercritical pitchfork bifurcation occurs in families of symmetric solutions of both the GS and DM types. A novel feature of the system is interplay between breakings of the PT and inter-core symmetries. Stability of symmetric GS and DM modes and their asymmetric counterparts, produced by SBBs of both types, is explored via the linear-stability analysis and simulations. It is found that the instability of PT-symmetric solutions takes place prior to the inter-core symmetry breaking. Surprisingly, stable inter-core-symmetric GS solutions may remain stable while the PT symmetry is broken. Fully asymmetric GS and DM solitons are only partially stable. Moreover, we construct symmetric and asymmetric GS solitons under the action of a pure imaginary localized potential, for which the SBB is subcritical. These results exhibit that stable solitons can still be found in dissipative systems. Finally, excitations of symmetric and asymmetric GS solitons are investigated by making the potential's parameters or the system's coupling constant functions, showing that GS solitons can be converted from an asymmetric shape onto a symmetric one under certain conditions. These results may pave the way for the study of linear and nonlinear phenomena in a dual-core planar waveguide with PT potential and related experimental designs.