What is the history between Doug and Rachel?

Rachel is the call girl/prostitute who was with Peter Russo in the first episode. If you haven't finished the season, then just keep watching.Spoilers below if you finished and missed it.

She is a cat's paw in Francis Underwood's plan. Doug was keeping Rachael around to be used against Peter Russo after they cleaned him up enough, for long enough, to run in the governors race. The plan was to use her to create a scandal at the 11th hour in the governor's race. The only way to keep governor's position for the party would be to have someone like the VP run at the last minute. After the VP ran and won, the VP position would be vacant leaving room for Francis Underwood to be nominated.

First of all, Zoredache's answer is perfectly correct about his main motivation for caring about Rachel. She was at first a facilitator in bringing down Russo, to be taken care of and prepared for her further employment in this case, and later simply a confidant in the whole conspiracy, to be hidden from the world and cared for to not talk about it. But you weren't too far away in assuming some kind of personal angle in their relationship, even if that only really manifests itself in the 2nd season. So there is a much more complex relationship going on, even if I'd still consider Zoredache's answer as the more straight-forward solution to your actual question, especially when concentrating more on the 1st season alone.

During his control visits of her at the place where they're hiding her, he gets increasingly obsessed with her. By forcing that much control over Rachel he somehow made himself dependent on her, or rather on the feeling of Rachel depending on him, which also motivates his tries in isolating her from everybody else (although, this is also motivated by hiding her and the secrets she knows, but not entirely).

He urges her to cut ties with Lisa, the girl she randomly met and whom she becomes friends with and later shares her appartment and a loving relationship with. It seems this isn't entirely motivated simply by taking care of Rachel not to be found, since Lisa likely doesn't present much of a danger in this regard. And in fact he temporarily gives in and lets her live with Rachel, maybe realizing that she is very important to her and not of an actual danger for Francis's plot, but only until discovering their actual sexual relationship, which makes him fall back to his strict control over her.

It already shows that there is more to his relationship to her when he repeatedly has Rachel read to him, especially A Tale of Two Cities, the book his mother read to him when he was young. This also shows that his increasing obsession is not merely of a sexual nature, but much deeper (or at least weirder), as Doug admits himself during an AA meeting:

I work hard, I keep things simple, I know what my priorities are. There's this...this person, she's not even in my life except on the edges, making things blurrier. It doesn't tempt me to drink, it's more like she feels what's like when I was drinking, when I couldn't get enough. No matter how many drinks I had I wanted another. I...I don't want to be with her. I mean I...I do, but...more like she's my daughter...my mother...I don't know, this is fucked up...

I should cut her off, the way I did with booze.

So there is a much more complex angle to Doug and Rachel's relationship, or at least to his feelings for her (her attitude towards him is rather grounded in fear, it seems). Likewise it is IMHO not entirely clear what Doug is really upto when he drives away with her at the end in S02E13. Was he really just relocating her, or was he about to more finally "cut her off" to get away from his addiction (which is at least what Rachel assumed when ultimately attacking him in supposed self-defense)?