1st you have to figure out what key your melody is in.(looks like Cmajor to me, could also be F) A good place to start would be to experiment with the chords of that particular key, but do let your ear be the final judge. There are no rules to composing, just guidelines to help us out when we get stuck in a rut.
CONNECTING CHORDS
Diminished 7th chords make for great connecting chords when used between two chords whose roots are a whole step apart.
EX; Amin7 D7 Gmaj7 G#dim7 back to Amin7
also try inserting a minor 7th chord a 4th away before going to the dominant.
EX; Bb9 Eb9 Edim7 Bb13 Cmin7 F9 Eb9
