I want to take a carry on, and another luggage that will go under the bus. However, I don't know will the count my backpack purse as luggage? And if so will I be allowed to take it and what's the fee?
It will not. If you can have it by your feet or on your lap or above your head (within reason), you can take it on board.
Source, have taken many, many Greyhound trips, including where I'd take a largeish bag and just put it by my feet for convenience. It was not small, but I was happy to rest my feet on it.