Who is responsible for overseeing the countrys finances?

In a democratic republic as an example, the executive branch and the legislative branch of government should be overseeing the nation's finances. Basically legislation involving programs and the money to fund them originates in the legislative branch. If this activity becomes a law by an act of signing it by a president for example, then we have dual oversights.

Let this be an example as it can become more complicated. Traditionally within the executive branch there is a department of the Treasury. The cabinet head and the chief executive of the government should be overseeing the proper expenditures of the government.

Also, since the legislative branch originated the financial law, it should have an "over sight committee" to also insure the finances are being spent properly.