We're just going to have to start this bit with a solid spoiler warning, so here you go: This article may contain spoilers not suitable for viewers who have not yet watched Star Wars: The Force Awakens. If you have not yet watched Star Wars: The Force Awakens, viewer discretion is advised. SPOILERS.
Now that that's out of the way, let's try to answer this question: Do Han Solo and General Leia Organa know who Rey is?
Well, first of all, we know it's more of a "Did" Han know, which is a heartbreaker in itself but, nonetheless, did he know? After a pair of viewings, I think so.
While Han acted like he had no idea who Rey was throughout the moments we were able to see him in, it sure does seem like he knows the scavenger girl.
First of all, Maz Katana didn't know Rey when she, Finn, and Han walked into her watering hole. When she and Finn left the table, Maz turned to Han and asked him about the girl but the scene took the audience elsewhere and led Rey down a staircase to Luke's lightsaber.
The next time Maz encountered Rey, she knew all about her. She suddenly knew she was from Jakku and, on top of that, knew she was waiting for something/one to come back. Where could this sudden knowledge have come from if not from a conversation with Han in the moments prior? Yes, Rey told Han that she needed to go back to Jakku, but the intimacy with which Maz addressed Rey over the situation seemed like much more was divulged than that.
At that point, Leia and Rey had never so much as seen a photograph of one another, let alone met. But after Han's death and Rey's return to the Resistance headquarter, Leia was waiting for her with open arms and a hug. No words were shared, no introduction, just an expected greeting and shared grief over Han's demise. Again, to play devil's advocate against ourselves here, they do share a connection in the Force, even if there's no familial connection, but this again had an intimacy to it that seemed stronger than simply "well, they're both Force-sensitive."
It appears that Rey is Luke's daughter, making Han and Leia her aunt and uncle. Sure, this begs the question of, "Who is her mother?" but that's for a whole different set of theorizing.
If Rey is indeed Luke's daughter, Han and Leia would almost certainly know about her. She's not much younger than Kylo Ren, maybe about 10 years (if we go by the ages of the actors), so by the time Ben Solo was training and turned to the dark side, probably in his teenage years, Rey would've been from the ages of 5 to 8 years old. That seems on track with when she was left behind on Jakku with Unkar.
But why would they keep it a secret?
Luke leaving his daughter behind seems like a bizarre thought but it could be a method of training for both or either of them. There's always the possibility that Luke genuinely had so much guilt and fear from his training of Ben Solo failing that he thought it best to exile himself from his own daughter's life and convinced his sister to play along for Rey's benefit. He also could've simply been trying to protect her from Kylo Ren, or keeping her somewhere until he found that first Jedi Temple people were so keen to briefly drop into conversation then never mention again.
It certainly did look like Rey had been aboard the Millennium Falcon before (she knew where the gunner position was, and how to fix it/how to spot modifications) and received a bit of Jedi training (how did she know how to do a Jedi mind trick with zero training?). This is best shown by her contrast with Finn. While we don't know much of his history aside from being plucked from his family and into the First Order and having fired blasters before, we can see that he is not very skilled with a lightsaber and knows nothing about the Millennium Falcon. Rey, on the other hand, suddenly remembered how to use the Force, handled a lightsaber impressively (though she was well-trained in hand-to-hand using the longer staff already, too), and knew the ins and outs of the Millennium Falcon better than Han himself.
When Rey and Han shared a moment outside of the ship, it looked like an indication of Han wanting to look after Rey, rather than simply offer her a job.
Most signs do point to Han and Leia being aware of Rey's existence and identity prior to meeting. The bigger question is, if Kylo Ren could "see" the ocean and island Luke was on in Rey's mind - how was it there in the first place? Was it a connection to Luke, a memory, or something else? But that's for another time.