The great thing about trying lots of Beatles songs on piano is that you can't predict in advance where they can take you. I've never thought of "Rain" as a "piano song," so I was pleasantly surprised when I set out to record it for my Complete Beatles Piano video series on YouTube.
The first thing you need to do in order to interpret Rain on piano is to thoroughly learn the melody and chords. Just sit down at the piano and kind of forget that you ever heard The Beatles recording of the song (don't worry, we'll come back to their recording later.) The Beatles' original versions of their songs are so wonderful and compelling that if we keep them foremost in our minds as we play them on piano, we risk having them fall flat. (And we might not even realize it ourselves, since we'll be mentally "hearing John, Paul, George, and Ringo in our imaginations, but our listeners may just be hearing a mildly pleasant piano version. We aim to go beyond "mildly pleasant!"
So take some time and simply listen to yourself play the chords and melody, even at a slow tempo, or out of tempo. Cleanse your palette. Get in to a musical flow, and experiment with a few different pianistic textures. Play it with emphatic chords. Mimic an acoustic guitar playing fingerstyle. Add a gospel touch. Semi-classical. Have fun and stretch a little bit. Go beyond your first impulse.
At this point, you can come back to The Beatles version without being hemmed in by it. You can now meet it halfway, by merging your style and the way The Beatles did it. (They tried all these things too, by the way, so you're in good company!)
I did all these things with their great song Rain, and found myself playing it in a very different way than I would have predicted in advance. It's in the key of G, and I got very interested in the F that Paul McCartney plays on his bass (it's the 7th of the G tonic chord). Since a G major scale with an F instead of F# gives us a G Mixolydian mode, I decided to stay on the G chord for a while and improvise using this scale. As a bonus, the G Mixolydian mode over this groove has a vaguely Middle Eastern sound, which The Beatles, particularly George Harrison, were getting more and more into at about the time they recorded Rain. The vocal harmonies during the chorus emphasize this feeling as well.
Have a listen to it here, and then try it yourself. Rain is a wonderful song to use for piano improvisation!