It is definitely possible to learn to play piano by yourself, but it would be better if someone could show you. As far as steps to teaching yourself, here is what I recommend:
1. Learn every key on the piano, every black and white key. What I mean by that is, if i say "Play the last/highest E for me" or "Play the second Bb from the bottom," you should be able to go to it and play it w/no hestitation.
2. Learn as much music theory as you can.
3. Learn all major scales and be able to play them using correct fingers. As far as sound and playing goes, there are 12 major scales. But, in actuallity there are 15 major scales. You don't believe me? Here they are:
C, C#, Db, D, Eb, E, F, F#, Gb, G, Ab, A, Bb, B, Cb (12 + 3 enharmonics)
4. Learn all basic types of chords and there inversions. They are the major, minor, augmented, and diminished chords. If I say to you "Play me a C minor chord in 2nd inversion," you should be able to first see it, then play w/out too much hestitation.
5. Learn about progressions and how to incoroporate them into your playing. I kid u not on this one, learning progressions will save you many hours on learning a song. I can pick up songs in like 5 minutes just using progressions.
6. Start learning everything above, especially the theory, in the Key of C. Once you got it down in C, don't stop there. Move on until you learn all the other keys as well.
If you need any specific help, ask me. That's what I'm here for.