I want to set a custom field on Vendor Bills called "Approved By", which captures the person who eventually approved the bill (through a custom workflow). Whats the best way to achieve this ?

It depends on how the Approval is occurring. If the Approval is within a workflow, the simplest way would be - in the state where the Workflow is setting the Vendor Bill to approved, add a Workflow action of 'Set Field Value' to 'Current User' in your Approved By field. This is of course assuming that the Approved By field is using List/Record 'Employee' and that an Employee is approving the Vendor Bill.

That works great @Tim Opsal, thanks!

Awesome! You bet!