As in most coils the primary circuit is running with voltages of 1kV - 22kV, the capacitances need to have dielectrics (insulation between their plates) capable of withstanding this sort of voltage. Usuall values of primary capacitor are 6nF - 10uF depending on coil size and primary voltage.
For maximum efficiency the capacitor needs to deliver all of its charge in a very short period of time, this is mostly restricted by its effictive series resistance(ESR) pulse currents of 400A are not unknown in Tesla coil primaries.
High Voltage Capacitor
MMC (Multi Manufactured
Difficualt & time consuming to make, Physically large for a given capacitance
Good ESR if constructed well, can be very cheap. e.g. beer bottle caps.