All we know about Capt. Greene is the snippets of MilitaryConversation removed from Melissa's memory banks and left to dry on the website by the TheSPDR.

Capt. Greene is female. She commands the Human starship for which Melissa was the AI. This was a light fast craft disguised as a pleasure yacht. The Captain had sealed orders for her mission.

During the voyage they were able to decrypt a Covenant transmission indicating the enemy were going to "glass" - bombard with energy weapons from orbit - a human colony planet named Troy. The Captain has family on Troy but knows hiding knowledge of the decryption may be more important than saving the colony planet. She forwards the decryption to the head office, presumably Admiral Standish.

She was killed by a accident with a bus. Survivor, McKaskill, announced to James James that there's been a "flu" going around the surviving crewmembers of the Apocalypso, like bus accidents. Most likely Capt. Greene's death wasn't an accident, but a conspired murder.