I have bought and downloaded both Hearthfire and Dragonborn from the Playstation Store and downloaded them, and when they finished downloading I installed them. However, upon loading the game and my character, the game was unchanged. I am unable to perform any of the new functions brought by the DLC.
What exactly have you tried? The new stuff will show up under the proper circumstances. For example, in Dragonborn, some cultists will come and try to kill you. It seems to me that if you're in a city and enter and leave a house, this should happen. For hearthfire you have to talk to a Jarl in one of the regions with a new home; I forget if there was something that prompted this as a quest.