Well, the Company funded its war expenses by raising India Bonds in England, from the banks in England and finally from the Presidency Banks in India. But how did its Indian adversaries do it?
Well, by the usual means: raising taxes from the population or by calling in their vassals who had to furnish horses and men. And in some cases they employed European mercenaries against their enemies. About the financial side you can consult Irfan Habib (though he deals mostly with precolonial Mughal finance) and Stig Förster about the creeping acquisition of territory by the EIC.