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I am learning RH chords, playing them along with LH chords. So right now, I'm playing 2 note RH chords. Suppose the measure calls for a C chord, and the melody note is an E note.
When playing below the melody note, do I play an E note + a B note from an E chord or do I play an E note + a C note from a C chord?
It sounds pretty good either way.
Taking your theory further, you could , while on that E melody note, play a G# as well. Now it stops sounding so good! The E melody note is contained, as Dom says, within the C maj chord. Playing an Emaj. chord wouldn't (usually) work as the G# is not in the Cmaj. set of notes - it's not diatonic. The B sometimes works, as it makes a Cmaj.7 chord, and of course, the B is diatonic in Cmaj.
The notes of a C major chord are C, E, and G. To fill out the chord if the melody has a E you could play the missing notes C and G. It's the simplest and most effect way of filling in the rest of the chord. There are other options though that I will explain.
If you are playing with others you have a lot more freedom in your voicing and as long as you are playing chord tones, but if you are playing alone there are certain tones you would need to imply a chord. To imply a C major chord you need at least the C and the E. You could leave out the 5th and just play C and E. You could also add the major 7th if you want too and play C and B, but you would technically be playing a CM7 chord instead of just a C major chord. You also could probably also play C and Bb to imply a C7 which depending on what you are playing may or may not make sense.
You have a lot of options and I suggest you play around with them to hear what sounds best to you.
If you're talking about keyboard playing, beware of playing close position chords in the LH. That register can get muddy, and it's where bass lines (or maybe counter-melodies), not chords, belong.
If you want a C chord, play the notes of a C chord. Colouring it by adding the 6th (A) or major 7th (B) is harmonically neutral. The minor 7th (Bb) turns it into something with a quite different function.
