Depends. To be a package car driver, you don't need a CDL anymore. Used to be a class C CDL, but they got rid of those.
If you put in enough time as a full time package car driver, they will pay for your class A CDL to be a feeder driver.
However, you posted in the freight section - so to be a driver for UPS freight, you need a Class A CDL with hazmat (its all on the UPSJOBS.COM website), and double endorsements to be hired on as a freight driver.