We consider how the Hamiltonian Quantum Computing scheme introduced in [arXiv:1509.01278] can be implemented using a 2D array of superconducting transmon qubits. We show how the scheme requires the engineering of strong attractive cross-Kerr and weak flip-flop or hopping interactions and we detail how this can be achieved. Our proposal uses a new electric circuit for obtaining the attractive cross-Kerr coupling between transmons via a dipole-like element. We discuss and numerically analyze the forward motion and execution of the computation and its dependence on coupling strengths and their variability. We extend [arXiv:1509.01278] by explicitly showing how to construct a direct Toffoli gate, thus establishing computational universality via the Hadamard and Toffoli gate or via controlled-Hadamard, Hadamard and CNOT.