When River Song first met the Doctor, it was at the end of her life. Since then, on each occasion, she meets the Doctor a earlier in her own life.
Why is this? Is there an in-universe explanation?
I would have thought that, since they are time travelers, they can meet in any order they like.
There isn't an in-universe explanation for this, as in universe, this isn't the case. From River's perspective, her final meeting with The Doctor was the adventure which ended in her death. But once she was imprisoned in Stormcage, The Doctor visited her often. Some of these visits are in the Doctor Who blu-ray extras. These visits would take place in between the episodes of Doctor Who that we see, and these visits would therefore prevent them from consistently meeting earlier and earlier in River's life.
The major adventures they have, we the audience see in episodes of Doctor Who, but these minor side adventures are merely referenced on screen. For the major adventures, we keep meeting River earlier in her timeline. However, those side adventures occur in The Doctor's timeline after the events of The Wedding of River Song. As such, in universe, River and the Doctor don't keep meeting earlier in her timeline.
River Song's Timeline isn't backwards (and neither is the Doctors). Their individual timelines move forward from birth to death (in the loosest sense of the word) but as time travellers they are free to visit any time and space in any order.
When they meet at specific historical events in time they have each come from different points in their individual timelines and it is these journeys that are in opposing orders.
The clearest demonstration of this I've seen is here:
Why are their journeys destined to bring them together in this order? Well that's just the unfortunate fate of Star crossed (Time crossed)? lovers.
Though it's speculation, I believe their meetings are Fixed Points in Time that is they can't meet in any order they want. With both of them being:
Time lords
It makes some sense that the way time moves between them is more rigid, else any
Time lord
meetings would be chaotic, as one could go back and prevent the meeting, and then the other could counter, and then the other.....
They are fated to meet in the order prescribed.
If you remember the whole incident about her being taken as a child, she is taken in "A Good Man Goes to War" and the next time we see her in the Doctor's timeline is the next episode, "Let's Kill Hitler". Even though she was only an infant at the first point, she met the Doctor for the first time there and the rest of the following events were in order for her.
(Her last meeting with the Doctor, when she "dies", also may not have been the last of her. There are stories about consciences being transferred from memory banks and the like to living bodies, like the Doctor's Daughter who was created from his DNA.)
So it isn't two straight timelines running parallel (one backward, one forward). "People assume that time is a strict progression of cause to effect, but actually, from a non-linear non-subjective viewpoint, it's more like a big bowl of wibbly wobbly timey wimey... stuff." Just remember that what we see onscreen is linear, but the Who-niverse is not.
If River never saved the Doctor in the library, then she most likely wouldn't have existed in the first place. By older River saving the 10th Doctor, it lets the 11th Doctor meet Amy and Rory, who in turn conceive a child on board the TARDIS while it is in the time vortex. This child is taken by the Silence, where she ends up as Amy and Rory's best friend. She then gets Amy and Rory to start dating, that will start the chain of events that leads to her birth. So basically that's why her life is fixed, because altering it could mean she never existed. It's backwards because it makes more sense that way.
