Romeo begins this monologue of love-stricken thought by comparing Juliet to the sun. He sees the light shining in her window as being the radiance and beauty of Juliet's presence, just as the sun sheds light on all its surroundings.
When the sun arises, the moon is no longer alight and visible. So, one interpretation of "Arise, fair sun, and kill the envious moon" can simply be recognizing that when the sun rises, the moon ceases to visibly exist.
Another interpretation could draw upon the traditional use of the moon as a symbol for Diana, the moon goddess. As Romeo continues, he explains that the moon is envious, "sick and pale with grief," because Juliet, the sun, is "far more fair than she (the moon)." Romeo ends this section of his thoughts, watching the light from Juliet's window, by encouraging Juliet to cease her symbolic relationship with the moon: "Be not her maid, since she is envious."