I think dogperson gives very wise answers to the question, which was a very good question by the way.
The interpretation, and the way we interpret, has to be very individual. As been said, first you must follow the composer's writings. But nothing is holy ... what matters when a pianist plays a piece, IMHO, is if the pianist in question HAS made some thinking or is just a "robot". This is here we often distinguish the experienced, skilled pianist from the beginner. I am often less impressed by young progidies than people around me, and this is because the musical interpretation. Playing the piano is certainly about a lot more than being able to pressing the right keys in super speed.
So if you really take your time and decide how you want to play each and every bar, then you should not worry if your interpretation is "wrong".
I am a fiction writer and I use this "talent" of mine when I interpret some pieces. My teacher encouraged me to do this. To some pieces I can think up whole scenes, with action and emotion, which I then try to "describe" with the music. Some people are perhaps more visually oriented and want to "paint a picture" in music.
Then, of course, you will get a lot of help from other people who already have made their analyses, and perhaps have a profound knowledge about the composer. For example, I recently learned that the third movement of sonata Patétique was originally intended as a piece for strings. What if you try to hear the violins and violas "behind" the notes? Wouldn't this affect your playing quite a lot?
Another example: Rondo C Major by Beethoven. I decided to interpret it as a piece written for an opera duo, where a woman is having a dialogue with her husband, an argument which ends where they have made some kind of agreement, even though their interpretation of this agreement are quite different ... (yes, I used my imagination here). And so I play with "his voice, her voice, his mood, her mood" in mind. This is something I must play in my head BEFORE I try to play it for real, as I am technically not that advanced - afterwards I often must conclude that it did not sound as I wanted to, and then I must work out solutions to get the sound I want, everywhere. But by then at least i have this musical image in my mind: this is what I want to achieve.
If you don't enjoy the idea of making up scenes in your head, then you can always just "hear" the music in your mind, over and over again. Sing it to yourself - if you don't have a good voice, then just sing it inside yourself, until you really know and really FEEL the music within you. There is you interpretation. Now, go and make the piano sound like that.