Here is schema, it is similar to one that I was using to drive motor Gray Tube Replication.
Capacitance and spark gap should be tuned in order to achieve desired frequency.
Capacitor next to the bulb actually is aluminum foil on top of plastic wrap around bulb base.
Metal plate at the bottom, need to think about that (Tesla was covering walls with special paint or metal sheets) may be variable capacitance.
Second coil - disassemble ignition coil and remove core. Separate coils by cutting HV wire connected with LV wire.