Chell's backstory has to be inferred from content in the game, as very little is made explicit. We know from Valve's pre Portal 2 Comic "Lab Rat" that Doug Ratmann placed her at the top of the testing queue because GladOS had previously rejected her as being unsuitable for testing due to excessive tenacity, but this tells us little about her origins.
Portal 2 spoilers follow:
One of the science exhibits at the "Bring your daughter to work" day is indicated as being by Chell. While everyone else seems to have made a potato clock, Chell created a mutant potato with accelerated growth. From this we can conclude (a) Chell is the daughter of an Aperture worker (b) she appears to be something of a scientific genius and (c) she has apparently aged somewhat by the time of the original Portal. (This aspect of the game is somewhat problematic - how has she had time to grow to adulthood if only woken for periodic tests? One can assume she's been in cryo for quite some time, but that doesn't seem consistent with Ratmann still being alive.)
More spoilers, including speculation based on in-game content.
GladOS finally letting go of Chell at the end of Portal 2 offers some clue as to Chell's parentage. As revealed by Portal 2 the personality upon which GladOS is based is that of Cave Johnson's secretary Caroline. Although GladOS claims to have erased Caroline at the end of Portal 2, it wouldn't be the first time she has lied. The Turret Opera in particular seems to be a song expressing deep motherly affection for Chell. It is likely that Chell's mother is actually Caroline. Her father remains a matter of pure speculation, but given Cave Johnson's apparent fondness for Caroline (as exhibited in the fact he wanted her to be used as the AI template), the idea that Cave Johnson himself is Chell's father remains a strong possibility. This is possibly backed up by Chell's scientific prowess, but it's never clear whether Cave Johnson is responsible for any of the actual Science at Apperture or whether he is a non-scientist CEO.
More background on Chell can be found here.