There are several options, depending upon when your flight lands.
If in the daytime and you are both frugal and adventurous, you can take a taxi to either Caribe Tours or Metro Bus where you can take a large, comfortable bus to Puerto Plata. You would need to take another taxi from the bus station in Puerto Plata to your final destination.
If arriving late at night, you can either take an airport taxi (approximately $100 US) or arrange a pick up with a local taxi service to take you to Puerto Plata. Using a local taxi (not airport) cost me 2500 pesos (slightly over $50 US) last June.
If you are arriving in the middle of the night, you can take a taxi to a hotel or cabaña, wait until morning, and then do the taxi/bus frugal trip to Puerto Plata. Of course, then you need to figure in the cost of the hotel, etc.