Hybrid circuit QED involves the study of coherent quantum physics in solid-state systems via their interactions with superconducting microwave circuits. Here we present a crucial step in the implementation of a hybrid superconducting qubit that employs a carbon nanotube as a Josephson junction. We realize the junction by contacting a carbon nanotube with a superconducting Pd/Al bilayer, and implement voltage tunability of the quantum circuit's frequency using a local electrostatic gate. We demonstrate a strong dispersive coupling to a coplanar waveguide resonator by investigating the gate-tunable resonator frequency. We extract qubit parameters from spectroscopy using dispersive readout and find qubit relaxation and coherence times in the range of $10$--$200\phantom{\rule{0.2em}{0ex}}\mathrm{ns}$.