She plans to talk her father into buying her a new car.
Literal
She [topic-は] father [object-を] persuade [then-て] new-car [object-を] buy-receive [intention-つもり] is.
説得して ('by persuading') + ~てもらう ('to receive the favor of') chains persuasion with the benefactive pattern. かって is an informal kana rendering of 買って (買う, 'to buy'). つもりです expresses intention.