Is there a simple assay to determine the elemental carbon content of biochar?
Assuming your charcoal is mostly:


water (evaporates)
carbon (burns)
ashes (does neither)


I would :


weigh a sample of your charcoal to get the mass $\ce m_0$
dry it in an oven at low-ish temperature (50°-80° or more, whatever doesn't burn your charcoal) for a few hours.
weigh it again to get $\ce m_1$
burn the sample in air as thoroughly as possible
weigh what's left to get $\ce m_2$


Now you can calculate :


mass of water (+ other volatiles) = $\ce m_0$ - $\ce m_1$
mass of carbon (+ other non-volatile burnables) = $\ce m_1$ - $\ce m_2$
mass of ashes = $\ce m_2$