River Song was born Melody Pond(-Williams) five thousand years after conception, and raised fifty years before. One might think with a birth like that, she'd have used up all of her near-paradoxes for a lifetime. Not true.
As it turned out, her life only got more complicated. At a young age, she met a man who'd known her all of the life she hadn't lived yet, who dropped probably more things about her future than she needed to know, and swept her off on adventures she'd never have seen before otherwise. As it happened, it was the man she'd been groomed from birth to kill...but that's another story entirely.
What's important is that she ended up rather tangled up with him instead. Of course, the details might not be the details you'd think. After she escaped from the various groups of people hoping to use her rather unique DNA for themselves, she found herself in the 51st century bereft of Time Travel. With a life so full of traveling up to that point, that simply wouldn't do for her. Using the scraps of technology from the shattered group who originally kidnapped her, she organized a group which would patrol time to ensure it maintained its proper course...and also provide a handy vehicle for her favorite hobby: archaeology.
As such groups tend to go, the Time Agency lasted only a short time before it unraveled. Then again, when the founder ends up imprisoned in Stormcage for murder and the rest of its agents end up little more than time traveling highwaymen, the only logical conclusion for it is a quick unraveling. It's rather worse when that murder is of the founder's own father, even if said founder wasn't aware of their parentage at the time.
Which brings the story back to its convoluted time line. Because River Song hadn't spent all that time behind a desk, filling out paperwork and only jumping back to spot artifacts before they became artifacts. No, her life was interspersed with meetings by the same man she met as a child. At first, he appeared as an enigmatic mentor, teaching her and showing her things she couldn't have dreamed of on her own, and telling her stories of parents she'd never known with a sad look in his eyes that told her not to inquire after them.
As she grew older and became capable of keeping up with him--even occasionally getting a leg up on him due to their curiously reversed timelines--they became more: they became very close and lasting friends. For the longest time, she never really knew why he'd taken an interest in her, but he shaped her youth, and soon became the highlight of her adult life.
Living in the 51st century, it didn't take long for her to realize her feelings for him moved on beyond friendship to infatuation, adoration, and even love. And being as clever and familiar with the Doctor as she was, it didn't take much longer than that to realize they were feelings he never quite returned. Oh, she knew he regarded her highly, that she might even be his closest, dearest friend in a very long time--and that said something in a life as long as his--but the love she felt for him never quite translated. She never let that knowledge keep her down, though, because no one ever said she needed his permission to love him.
Of course, the fact that every time she met him she felt a little bit more and he a little bit less...that did get to her, every now and then. When she would look forward to seeing him again, only to see that look of confusion and curiosity growing more every time. Knowing their timeline was back-to-front (with brief brilliant moments of exception) only made that worse.
From childhood, however, she built up a rapport with the Doctor. Knowing their timelines were backward, and knowing how difficult it would be to keep their own past out of the other's knowledge of their future, they started a game. They wrote in journals to keep their timelines straight, and when they were sure of what the other knew and didn't know, they'd liberally make things up about the 'future'. They made it a challenge to see how outrageous a lie they could pass off as truth if they blended it in with more believable truths. In the beginning, he fooled her more often than not. Later on, she fooled him.
Their adventures ranged all across the spectrum, and sometimes their 'adventures' would look more like dates to the unobservant eye. Maybe they were. When he took her out ice-skating for her birthday, or when they'd happen across each other on merchant planets or space bars where she wouldn't have expected to see him, and they'd spend the night drinking and laughing. There was quite a bit of the drinking and laughing. It could be he indulged her feelings, such as they were, but she preferred to believe adventures and birthdays and drinking nights and the occasional snog after a harrowing adventure or one too many drinks was just their unique life.
A life she found herself holding on to more tightly every time he came to visit or called her out of Stormcage, as it seemed their time quickly sped through her fingers, and undying trust became suspicion. At times she despaired the thought that she would never again have that rapport with the Doctor, at others she pretended she still did and played the game with him solo. Often, she dreaded the moment she'd see her Doctor and he'd see a stranger.
Something happened near the end of her stay in Stormcage that changed everything. The isolated cases where she'd see him in a random order became more prominent, and the cases in which she'd be following their inexorable backward timeline became isolated. She never knew quite what happened to cause it, but it gave her a second chance at her lifetime, as she realized she still had countless adventures to have. She sought a pardon for her unwitting crime, and she stepped back out into the universe with the promise of becoming a professor.
The game was on again in earnest, and her diary got quite the workout, trying to figure out wherever they were in each other's timelines again. She reclaimed her life, and they ran across the universe once more. In one of the more remarkable adventures, she ended up with his childhood memories dumped in her head. For the most part, it meant teasing him about failing tests and spending his time building a sonic screwdriver instead of taking the pretty girl in class to the dance. Remarkably, it also meant learning his true name--which wasn't nearly as impressive as she would have expected.
In the end, when the day finally came that she spotted another face of the Doctor and he stared at her blankly, she wasn't even expecting it. In fact, at first she couldn't believe it. When she realized it was true, she did what she always had: she played the game.
By the end she realized the truth. The reason for the game they'd played, why their life had been so entangled, and what she'd always been meant to do. She'd been born to kill him, but ultimately she'd lived to save him. She chose her fate and her lifetime gladly.