Are you aware you are looking at around 8 hours of driving? You should look into taking a plane. Otherwise, you would be best off taking the Expreso Bavaro bus to Santo Domingo (less than 4 hours) and then the Metro or Caribe Tours Bus to Puerto Plata. From Puerto Plata a taxi to the airport.
I would only recommend driving if you can split this into two days because to do it in one day would be very difficult and even stressful for a Dominican.
Without going into much detail, to drive to Puerto Plata from Punta Cana you first need to get to Santo Domingo. So, from PUJ airport you get to Veron. From Veron to La Otra Banda and then to Higuey. From Higuey you take the highway that heads West to Santo Domingo. You do not want to go to San Pedro de Macoris because that root is slower. Probably you could find someone who is leaving from the PUJ airport and you could tail them into Santo Domingo.
Once you get to Santo Domingo you need to take the Elevados after the Ozama Bridge and then get off on the right detour right after the detour of Dr. Delgado (think it is V Centenario, but not sure). Then you follow that road to where it ends which is Leopoldo Navarro and take a right (you only can go right) and then at the stop light you take a left which puts you on John F. Kennedy avenue that goes west and then northwest and will take you to Santiago by going straight all the time. Once you get to Santiago you need to take Estrella Sadhala avenue that will take you to the Navarette road and then you need to stay on that until you see the road that says Puerto Plata (small sign so watch for it). That will take you straight to Puerto Plata and when the road stops going north you take a right and that will take you straight to the Gregorio Luperon International Airport (POP), the Puerto Plata airport.
My suggestion is that you ask when you think you are wrong, and ask when you think you are right all the way, going from point to point.
Good luck!