The first thing you must learn when fingerpicking a chord is the bass line. The standard bass line for an E chord alternates between the open E string (sixth string) and the E note on the second fret of the fourth string.
While the right hand thumb is playing the standard bass line, the right hand fingers are playing a melody on the first three string. The notes which are played are taken from the E Major scale and the E Blues scale.
Example One is played by using an optional E Major fingering. A hammer is used on the third string.
Example Two uses two extra notes from the E Sixth and the E Seventh chords.
Example Three uses the second and third frets of the first string to expand the melody as well as another hammer on the second string.
Example Four shows how to bend the note on the third fret of the first and second string as an extra effect.