I heard that Cave Johnson & Caroline are Chell's parents but with their love of science, didn't have time to raise a child so chell ended up being adopted by employee Doug Rattman so that Cave & Caroline could still keep her close by! (Although there is slight credit to the possibility that she may be Gordon & Alex's biological child as she does seem to have some ethnic colour to her which is more fitting to Alex than Cave or Caroline).
It has however been said that GLaDOS' AI was completed and the untested AI was activated during the company's first ever bring-your-daughter-to-work day in May 2000. Immediately after activation, the facility was flooded with deadly neurotoxin by the AI. Events of the first Half-Life game occur shortly thereafter, presumably leaving the facility forgotten by the outside world due to apocalyptic happenings.
Wolpaw, in describing the ending of Portal 2, affirmed that the Combine invasion from Half-Life 2 occurred during Portal's events so it ‘seems’ unlikely that Chell is Gordon's & Alex's child without some kind of ‘timeplay’ occuring. (Possible HL3 spoiler?). Aperture Science does exist in the Half-Life universe, with the events of Portal occurring sometime between the story of Half-Life and Half-Life 2. According to The Final Hours of Portal 2, the year is established to be "somewhere in 2010" — ten years after its abandonment.
Aperture Science Inc. is also mentioned during Half-Life 2: Episode Two, in which Aperture's icebreaker ship Borealis is said to have disappeared under mysterious circumstances, along with the crew and part of its drydock. During its development, Half-Life 2 featured a chapter set on the Borealis, but this was abandoned and removed before release.
Gordon does however get to travel through time by G Man (he was held in stasis outside of time by him at the end of Half Life until needed for the events of Half Life 2. (Whether or not G Man can make Gordon go back through time as G Man seems to be able to do has yet to be seen).
Anyway, Cave contracts mercury poisoning during early (whilst in the 1st of a later 3 additional tiers of Aperture building complexes) experiments and eventually dies before his wish of having his consciousness implanted into AI (you can hear him talk about the disease in one of the office speakers in the old Aperture complex) although Caroline's eventually was!
After the hostile takeover by GLaDOS where all the employees are killed ,exept for Rattman who soon becomes schitzophrenic after hiding from GLaDOS, GLaDOS places Chell - who is unconscious - into a relaxation vault and after some time, Chell awakes to hear GLaDOS' voice telling her she is now a test subject. Cue the first Portal game.
After defeating GLaDOS, there is a huge explosion due to GLaDOS' destroyed generators creating a vortex. You then see Chell outside the facility lying face down unconscious. A patch was released prior to Portal 2 showing Chell being dragged away by an unseen robot which says “Thank you for assuming the party escort submission position” (which GLaDOS previously pleaded Chell to do in order to be taken to get a ‘reward’ when Chell escaped the incinerator ‘test’).
Doug Rattman witnesses the apparent destruction of GLaDOS, and later, after escaping from the facility himself, witnesses the recapturing of Chell by the Party Escort Robot. Following the robot, he finds that Chell has been placed into an Extended Relaxation Vault. So he heads back in and attempts to rescue her. However, as a result of the explosion at the conclusion of Portal, all of the relaxation vaults' life support systems are offline. Doug patches Chell's vault onto a reserve grid (as he was unable to open the vault), allowing Chell to survive the years prior to her reawakening in Portal 2.
At the end of Portal 2, GLaDOS saves Chell from space because of the compassion she has for her whilst still being partly Caroline, ie. Chell's mother. After GLaDOS deletes Caroline it still lets Chell live and leave knowing that she is just too dangerous to have around (and as she just can't seem to kill her). A part of me thinks that a part of Caroline still exists in GLaDOS or that she didn't really delete her but that's just an incomplete opinion (must do more testing;)
Maybe if there's a third Portal we'll find out, though I'm not expecting there to be a third part of the Portal story seeing as GLaDOS now has ATLAS & P-BODY, but who knows;)