A few things to note about playing on both PC and PS4:
1) As others have said, yes you need to buy a copy of the game (+ expansions) for each system you want to play on.
2) You do NOT need to pay multiple subscription fees. And any character on your account on the PS4 will be available on the PC as long as you activate the PC key properly. Make sure you do not set it up as a new subscription but instead add it to the current one.
3) Any pre-order, CE, bonus keys you've claimed will also be available on the other system. So no need for extras there.
4) May be a good time to wait for that complete edition coming out with the expansion so you can get Base + Heavensward + Stormblood all together for PC.