How about we tell the story as we currently know it through Melody/River’s POV?
Born on Demon’s Run and quickly whisked away with a ganger left behind.
Then raised for several years by the Silents, conditioned to kill The Doctor. There were pictures in the orphanage of presumed baby Melody with Amy as a baby. Not sure if they fit in anywhere though. United States space program born specifically to produce the astronaut suit for her to use, but she somehow tore her way out of it.
She then wanders the streets for a while, regenerates and becomes Mels, who finds her way, or is found and is placed, to be childhood BFF with her parents, Amy and Rory, and hence to meet up with The Doctor. Along that way feels she is a happy psychopath; states that along that time she wanted to marry him in her fantasies.
First we see/hear of this Mels version is as she squeals up in manner that makes us expect River.
Regenerates into the form of River Song and poisons the Doctor, toys with Nazis, watches the Doctor die but as he dies is apparently somehow moved that he still cares about trying to save her childhood BFFs who she knows, and always has known, are her parents, and impressed by how much he cares about this River woman. Those things are enough for her to go into the Tardis and the Tardis teaches her how to fly her; she saves her parents/childhood BFFs. The Tardis has now communicated with her, and she is the Tardis’s daughter too, in a sense. After that The Doctor asks her to find River Song and tell her something whispered and when she then realizes she IS River, and it is clear that she HAS successfully completed her programming to kill him, she brings him back to life with some sacrifice.
Somehow she finds her way, without a Tardis, to a future century to become an archeologist, figuring that she can use that position to get the information she needs to find him again.
{Missing events, possibly to be filled in in the last episode of the year, “The Wedding of River Song”}[spoiler]In the most recent issue of the Radio Times, Steven Moffat has summarised the episode:
And here it is - the episode with all the answers, as the doctor journeys to Lake SIlencio for his date with destiny. While I was writing this, my wife asked me, “what happens?” and when I frowned, and pondered my answer, she hurriedly added “In one sentence. Quite a short one.” I thought a moment and said, “The Doctor goes to his certain death - and it doesn’t work out as well as he expected”.
So welcome to what is, without doubt or serious competition, the wedding of the year! Early reports indicate that it could make On Her Majesty’s Secret Service look like a fairly successful honeymoon…
Also, the magazine revealed some of the cast list for episode 13, which includes Simon Callow (who played Charles Dickens in The Unquiet Dead), as well as Frances Barber as Madame Kovarian and Simon Fisher-Becker as Dorium Maldovar. Charles Dickens also makes an appearance.
[/spoiler]
Presumably she, inside the astronaut suit, kills a Matt Smith version Doctor who is 200 years older, and escapes while being shot at by an older version of herself. Presumably that the is the crime that she is then in prison for (when she is not escaped to be with The Doctor). We do not know how she goes from sacrificing her own regenerations to killing this version of The Doctor. But she feels she deserves to be in prison. Where that fits within the missing events is not known.
At some point after that she shows up at Demon’s Run after the battle and reveals that she is baby Melody now grown to The Doctor and to her parents. She does not reveal that she was also their childhood BFF. Reprimands The Doctor over what he has become such that he is viewed as a great warrior-destroyer.
She has adventures with a version of/The Doctor: Jimmy the Fish, picnics, etc … and meets her father Rory again, who she knows at this point does not know he is her father, and tells him she can’t come to help the battle at Demon Run.
She is summoned to be there to witness the person in the astronaut suit kill The Doctor and seems honestly surprised to see it happening. Acts and behaves as if she does not know who the girl in the space suit is. Is hunted down on behalf of The Silents and rescues Amy from them. Is part of their presumed undoing. Along the way states how sad she is to know that he knows her less and that someday he won’t know her at all, that that may kill her. Ends with her sadly having her last kiss with him.
Delivers a message to The Doctor in the form of a painting by Vincent and helps reboot the Universe in The Pandorica episodes. Is somehow outside of the change of time enough that she still has her book and can tell him that she will know who she is soon. Has been dealing with her parents not knowing who she is for a while now and is well practiced at not giving that away.
Is released from prison under guard to deal with the Weeping Angels and is coy about whether or not she should be trusted. Reveals that she murdered the best man she’s ever known. Is seeing her mother for what she knows is the last time but plays it very cool.
Is released from prison to investigate the library and comes across a Doctor who is not her Doctor and is very sad to realize that he does not know her. Knows what will happen to Donna and is sad about it for her but won’t reveal. Does reveal that she knows his name. She sacrifices herself to save him but her essence is saved in the matrix of the Library.
So what can be in the missing parts to make a cohesive story from her POV?