I can't speak for civil expenses but I can tell you about the military. Starting with Augustus, the emperor paid nearly all the wages, donatives, and pension at the end. Augustus, when he become emperor he wanted the soldiers to be loyal to him and not to the state so he used 170,000,000 sesterti to start a military treasury, which he used to pay the soldiers of the civil war and the soldiers after.

During emperors birthdays, major victories and other celebratory event, emperors would usually give their troops donatives, which are just a cash bonus.

Also when Roman legionnaires retired, it was the emperor who was giving them their severance package.

Sources:

The republican roman army. Sage, Michael M.

The Roman Army. Southern, Pat

While not directly on point, there is some authority for the view that there was a legal distinction between the Roman state and its treasury. The state could not be sued, but the treasury could.

"[A]s the Emperor's power increased and that of the populus diminished, the Imperial treasury, the fiscus, was created, and the question lost importance, for the fiscus, the State in its property relations, could be sued."

While this may have been merely a legal distinction, it supports the idea that the Emperor did not treat the treasury as his personal funds.

Source: Edwin M. Borchard, Governmental Responsibility in Tort, VI, 36 Yale L.J. 1, 5 (1926).