A bird is of all beings
Likest to the dawn
Not 'like' but 'likest'—the most similar. Dickinson is ranking all creatures and putting birds at the top of the dawn-resemblance scale.
General Heavens upon
Strange syntax—'put afloat / The general Heavens upon' inverts normal word order. The bird doesn't float in heaven; it floats heaven itself, as if the sky is a light object.
Measures with the clouds
'Measures' suggests both comparing size and dancing in rhythm. The bird uses clouds as a yardstick or dance partner.
Wake of music
A 'wake' is the trail behind a boat. Birdsong becomes a physical trail left by their feet, not their beaks—Dickinson swaps the body part that makes sound.
Should the Dawn emit
'Should' means 'if only'—she's wishing dawn made sound the way birds do. The bird improves on dawn by adding what dawn lacks.