multiple regression analysis to examine this model

you can use structural equation modelling for this purpose

A naive way of doing it would be to regress A~B.

Then repeat by including C in the model A~B +C. This would give you an indication of how much of the relationship between A and B is mediated through C. But this method may produce bias results.

In the real setting, you will have other variables that could potentially affect C like unmeasured mediator-outcome confounding variable. There might be an interaction between exposure and mediator, and presence of intermediate confounding. Regression may not be sufficient in this instance.

You can consider using the following methods in causal inference framework:

a) Inverse probability of treatment weighting

b) G-estimator

c) G-Computation

Thanks alot for the valuable answers of all.