How do I become a paid caregiver for my father?

Mythical777, welcome to the forum. Please note that the vast majority of grown children who are caring for a parent or grandparent do not get paid..... unless your parent can pay you from their savings account. If yes, you would need to create a Caregiving Contract, and have payroll taxes taken out.

Otherwise, see if your Father can qualify for Medicaid [different from Medicare] and see if your State offers some type of payment. Don't be surprised if the payment is minimum wage with only a few hours per week. Note that Medicaid is funded by us taxpayers.